Chapter 6

Find \(\frac{d y}{d x} .\) $$ y=\int_{0}^{x} 2 u^{3} d u $$

Approximate the area under the parabola \(y=x^{2}\) from 0 to 1 . using four equal subintervals with left endpoints.

Find \(\frac{d y}{d x} .\) $$ y=\int_{0}^{x}\left(1-\frac{u^{4}}{2}\right) d u $$

Find the areas of the regions bounded by the lines and curves. \(y=2 x^{2}-1, y=2-x^{4}\)

Approximate the area under the parabola \(y=x^{2}\) from 0 to 1 , using five equal subintervals with midpoints.

Find \(\frac{d y}{d x} .\) $$ y=\int_{0}^{x}\left(4 u^{2}-3\right) d u $$

Find the areas of the regions bounded by the lines and curves. \(y=e^{x / 2}, y=-x, x=0, x=2\)

Approximate the area under the parabola \(y=x^{2}\) from 0 to 1 . using four equal subintervals with right endpoints.

Find \(\frac{d y}{d x} .\) $$ y=\int_{0}^{x}\left(3+u^{4}\right) d u $$

Find the areas of the regions bounded by the lines and curves. \(y=\cos x, y=1, x=0, x=\frac{\pi}{2}\)

Approximate the area under the parabola \(y=1-x^{2}\) from 0 to 1 , using five equal subintervals with (a) left endpoints and (b) right endpoints.

Find \(\frac{d y}{d x} .\) $$ y=\int_{0}^{x} \sqrt{1+2 u} d u, x>0 $$

Find the areas of the regions bounded by the lines and curves. \(y=x^{2}+1, y=4 x-2\) (in the first quadrant)

Write each sum in expanded form. $$ \sum_{k=1}^{4} \sqrt{k} $$

Find \(\frac{d y}{d x} .\) $$ y=\int_{0}^{x} \sqrt{1+u^{2}} d u, x>0 $$

Find the areas of the regions bounded by the lines and curves. \(y=x^{2}, y=2-x, y=0\) (in the first quadrant)

Write each sum in expanded form. $$ \sum_{k=3}^{5}(k-1)^{2} $$

Find \(\frac{d y}{d x} .\) $$ y=\int_{0}^{x} \sqrt{1+\sin ^{2} u} d u, x>0 $$

Find the areas of the regions bounded by the lines and curves. \(y=x^{2}, y=\frac{1}{x}, y=4\) (in the first quadrant)

Write each sum in expanded form. $$ \sum_{k=2}^{6} 3^{k} $$

Find \(\frac{d y}{d x} .\) $$ y=\int_{0}^{x} \sqrt{2+\csc ^{2} u} d u, x>0 $$

Find the areas of the regions bounded by the lines and curves. \(y=\sin x, y=\cos x\) from \(x=0\) to \(x=\frac{\pi}{4}\)

Write each sum in expanded form. $$ \sum_{k=1}^{3} \frac{k^{2}}{k^{2}+1} $$

Find \(\frac{d y}{d x} .\) $$ y=\int_{3}^{x} u e^{4 u} d u $$

Find the areas of the regions bounded by the lines and curves. \(y=\sin x, y=1\) from \(x=0\) to \(x=\frac{\pi}{2}\)

Write each sum in expanded form. $$ \sum_{k=0}^{3}(x+1)^{k} $$

Find \(\frac{d y}{d x} .\) $$ y=\int_{1}^{x} u e^{-u^{2}} d u $$

Find the areas of the regions bounded by the lines and curves. \(y=x^{2}, y=(x-2)^{2}, y=0\) from \(x=0\) to \(x=2\)

Find \(\frac{d y}{d x} .\) $$ y=\int_{-2}^{x} \frac{1}{u+3} d u, x>-2 $$

Find the areas of the regions bounded by the lines and curves. \(y=x^{2}, y=x^{3}\) from \(x=0\) to \(x=2\)

Write each sum in expanded form. $$ \sum_{k=0}^{3}(-1)^{k+1} $$

Find \(\frac{d y}{d x} .\) $$ y=\int_{-1}^{x} \frac{2}{2+u^{2}} d u $$

Find the areas of the regions bounded by the lines and curves. \(y=e^{-x}, y=x+1\) from \(x=-1\) to \(x=1\)

Write each sum in expanded form. $$ \sum_{k=1}^{n} f\left(c_{k}\right) \Delta x_{k} $$

Find \(\frac{d y}{d x} .\) $$ y=\int_{\pi / 2}^{x} \sin \left(u^{2}+1\right) d u $$

In Problems 13-16, find the areas of the regions bounded by the lines and curves by expressing \(x\) as a function of \(y\) and integrating with respect to \(y .\) \(y=x^{2}, y=(x-2)^{2}, y=0\) from \(x=0\) to \(x=2\)

Write each sum in expanded form. $$ \sum_{k=1}^{n}\left(\frac{k}{n}\right)^{2} \frac{1}{n} $$

Find \(\frac{d y}{d x} .\) $$ y=\int_{\pi / 4}^{x} \cos ^{2}(u-3) d u $$

Find the areas of the regions bounded by the lines and curves by expressing \(x\) as a function of \(y\) and integrating with respect to \(y .\) \(y=x, y x=1, y=\frac{1}{2}\) (in the first quadrant)

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{3 x}\left(1+t^{2}\right) d t $$

Find the areas of the regions bounded by the lines and curves by expressing \(x\) as a function of \(y\) and integrating with respect to \(y .\) \(x=(y-1)^{2}+3, x=1-(y-1)^{2}\) from \(y=0\) to \(y=2\) (in the first quadrant)

Write each sum in sigma notation. $$ 2+4+6+8+\cdots+2 n $$

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{2 x-1}\left(t^{3}-2\right) d t $$

Find the areas of the regions bounded by the lines and curves by expressing \(x\) as a function of \(y\) and integrating with respect to \(y .\) \(x=(y-1)^{2}-1, x=(y-1)^{2}+1\) from \(y=0\) to \(y=2\)

Write each sum in sigma notation. $$ \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}} $$

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{1-4 x}\left(2 t^{2}+1\right) d t $$

Consider a population whose size at time \(t\) is \(N(t)\) and whose dynamics are given by the initial-value problem $$ \frac{d N}{d t}=e^{-t} $$ with \(N(0)=100\). (a) Find \(N(t)\) by solving the initial-value problem. (b) Compute the cumulative change in population size between \(t=0\) and \(t=5\) (c) Express the cumulative change in population size between time 0 and time \(t\) as an integral. Give a geometric interpretation of this quantity.

Write each sum in sigma notation. $$ \ln 2+\ln 3+\ln 4+\ln 5 $$

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{3 x+2}\left(1+t^{3}\right) d t $$

Suppose that a change in biomass \(B(t)\) at time \(t\) during the interval \([0,12]\) follows the equation $$ \frac{d}{d t} B(t)=\cos \left(\frac{\pi}{6} t\right) $$ for \(0 \leq t \leq 12\). (a) Graph \(\frac{d B}{d t}\) as a function of \(t\). (b) Suppose that \(B(0)=B_{0}\). Express the cumulative change in biomass during the interval \([0, t]\) as an integral. Give a geometric interpretation. What is the value of the biomass at the end of the interval \([0,12]\) compared with the value at time \(0 ?\) How are these two quantities related to the cumulative change in the biomass during the interval \([0,12] ?\)