Chapter 6

For the following exercises, determine the area of the region between the two curves in the given figure by integrating over the \(x\) -axis. \(y=x^{2}-3\) and \(y=1\).

For the following exercises, determine the area of the region between the two curves in the given figure by integrating over the \(x\) -axis. \(y=x^{2}\) and \(y=3 x+4\).

For the following exercises, split the region between the two curves into two smaller regions, then determine the area by integrating over the \(x\) -axis. Note that you will have two integrals to solve. $$ y=x^{3} \text { and } y=x^{2}+x $$

For the following exercises, split the region between the two curves into two smaller regions, then determine the area by integrating over the \(x\) -axis. Note that you will have two integrals to solve. $$ y=\cos \theta \text { and } y=0.5, \text { for } 0 \leq \theta \leq \pi $$

For the following exercises, determine the area of the region between the two curves by integrating over the \(y\) -axis. $$ x=y^{2} \text { and } x=9 $$

For the following exercises, determine the area of the region between the two curves by integrating over the \(y\) -axis. $$ y=x \text { and } x=y^{2} $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis. $$ y=x^{2} \text { and } y=-x^{2}+18 x $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis. $$ y=\frac{1}{x}, y=\frac{1}{x^{2}}, \text { and } x=3 $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis. $$ y=\cos x \text { and } y=\cos ^{2} x \text { on } x=[-\pi, \pi] $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis. $$ y=e^{x}, y=e^{2 x-1}, \text { and } x=0 $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis. $$ y=e^{x}, y=e^{-x}, x=-1 \text { and } x=1 $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis. $$ y=e, y=e^{x}, \text { and } y=e^{-x} $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis. $$ y=|x| \text { and } y=x^{2} $$

For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area. $$ y=\sin (\pi x), y=2 x, \text { and } x>0 $$

For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area. $$ y=12-x, y=\sqrt{x}, \text { and } y=1 $$

For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area. $$ y=\sin x \text { and } y=\cos x \text { over } x=[-\pi, \pi] $$

For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area. $$ y=x^{3} \text { and } y=x^{2}-2 x \text { over } x=[-1,1] $$

For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area. $$ y=x^{2}+9 \text { and } y=10+2 x \text { over } x=[-1,3] $$

For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area. $$ y=x^{3}+3 x \text { and } y=4 x $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(y\) -axis. \(x=y^{3}\) and \(x=3 y-2\).

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(y\) -axis. $$ x=2 y \text { and } x=y^{3}-y $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(y\) -axis. $$ x=-3+y^{2} \text { and } x=y-y^{2} $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(y\) -axis. $$ y^{2}=x \text { and } x=y+2 $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(y\) -axis. $$ x=|y| \text { and } 2 x=-y^{2}+2 $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(y\) -axis. $$ x=\sin y, x=\cos (2 y), y=\pi / 2, \text { and } y=-\pi / 2 $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis or \(y\) -axis, whichever seems more convenient. $$ x=y^{4} \text { and } x=y^{5} $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis or \(y\) -axis, whichever seems more convenient. $$ y=x e^{x}, y=e^{x}, x=0, \text { and } x=1 $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis or \(y\) -axis, whichever seems more convenient. $$ y=x^{6} \text { and } y=x^{4} $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis or \(y\) -axis, whichever seems more convenient. $$ x=y^{3}+2 y^{2}+1 \text { and } x=-y^{2}+1 $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis or \(y\) -axis, whichever seems more convenient. $$ y=|x| \text { and } y=x^{2}-1 $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis or \(y\) -axis, whichever seems more convenient. $$ y=4-3 x \text { and } y=\frac{1}{x} $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis or \(y\) -axis, whichever seems more convenient. $$ y=\sin x, x=-\pi / 6, x=\pi / 6, \text { and } y=\cos ^{3} x $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis or \(y\) -axis, whichever seems more convenient. $$ y=x^{2}-3 x+2 \text { and } y=x^{3}-2 x^{2}-x+2 $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis or \(y\) -axis, whichever seems more convenient. $$ y=2 \cos ^{3}(3 x), y=-1, x=\frac{\pi}{4}, \text { and } x=-\frac{\pi}{4} $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis or \(y\) -axis, whichever seems more convenient. $$ y+y^{3}=x \text { and } 2 y=x $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis or \(y\) -axis, whichever seems more convenient. $$ y=\sqrt{1-x^{2}} \text { and } y=x^{2}-1 $$

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the \(x\) -axis or \(y\) -axis, whichever seems more convenient. $$ y=\cos ^{-1} x, y=\sin ^{-1} x, x=-1, \text { and } x=1 $$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. $$ [T]x=e^{y} \text { and } y=x-2 $$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. $$ [T]y=x^{2} \text { and } y=\sqrt{1-x^{2}} $$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. $$ [T]y=3 x^{2}+8 x+9 \text { and } 3 y=x+24 $$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. $$ [T]x=\sqrt{4-y^{2}} \text { and } y^{2}=1+x^{2} $$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. $$ [T]x^{2}=y^{3} \text { and } x=3 y $$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. $$ [T]y=\sin ^{3} x+2, y=\tan x, x=-1.5, \text { and } x=1.5 $$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. $$ [T]y=\sqrt{1-x^{2}} \text { and } y^{2}=x^{2} $$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. $$ [T]y=\sqrt{1-x^{2}} \text { and } y=x^{2}+2 x+1 $$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. $$ [T]x=4-y^{2} \text { and } x=1+3 y+y^{2} $$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. $$ [T]y=\cos x, y=e^{x}, x=-\pi, \text { and } x=0 $$

The largest triangle with a base on the \(x\) -axis that fits inside the upper half of the unit circle \(y^{2}+x^{2}=1\) is given by \(y=1+x\) and \(y=1-x\). See the following figure. What is the area inside the semicircle but outside the triangle?

A factory selling cell phones has a marginal cost function \(C(x)=0.01 x^{2}-3 x+229,\) where \(x\) represents the number of cell phones, and a marginal revenue function given by \(R(x)=429-2 x .\) Find the area between the graphs of these curves and \(x=0 .\) What does this area represent?

The tortoise versus the hare: The speed of the hare is given by the sinusoidal function \(H(t)=1-\cos ((\pi t) / 2)\) whereas the speed of the tortoise is \(T(t)=(1 / 2) \tan ^{-1}(t / 4), \quad\) where \(t\) is time measured in hours and the speed is measured in miles per hour. Find the area between the curves from time \(t=0\) to the first time after one hour when the tortoise and hare are traveling at the same speed. What does it represent? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.