Chapter 5

Explain why, if \(f(b) \geq 0\) and \(f\) is decreasing on \([a, b],\) that the left endpoint estimate is an upper bound for the area below the graph of \(f\) on \([a, b]\)

Explain why, if \(f\) is increasing on \([a, b],\) the error between either \(L_{N}\) or \(R_{N}\) and the area \(A\) below the graph of \(f\) is at most \((b-a) \frac{f(b)-f(a)}{N}\)

A unit circle is made up of n wedges equivalent to the inner wedge in the figure. The base of the inner triangle is 1 unit and its height is \(\sin \left(\frac{\pi}{n}\right) .\) The base of the outer triangle is \( B=\cos \left(\frac{\pi}{n}\right)+\sin \left(\frac{\pi}{n}\right) \tan \left(\frac{\pi}{n}\right) \) and the height is \(H=B \sin \left(\frac{2 \pi}{n}\right)\). Use this information to argue that the area of a unit circle is equal to \(\pi .\)

In the following exercises, express the limits as integrals. $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(x_{i}^{*}\right) \Delta x \text { over }[1,3]$$

In the following exercises, express the limits as integrals. $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(5\left(x_{i}^{*}\right)^{2}-3\left(x_{i}^{*}\right)^{3}\right) \Delta x \text { over }[0,2]$$

In the following exercises, express the limits as integrals. $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \sin ^{2}\left(2 \pi x_{i}^{*}\right) \Delta x \text { over }[0,1]$$

In the following exercises, express the limits as integrals. $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \cos ^{2}\left(2 \pi x_{i}^{*}\right) \Delta x \text { over }[0,1] $$

In the following exercises, given \(L_{n}\) or \(R_{n}\) as indicated,express their limits as \(n \rightarrow \infty\) as definite integrals, identifying the correct intervals. $$L_{n}=\frac{1}{n} \sum_{i=1}^{n} \frac{i-1}{n}$$

In the following exercises, given \(L_{n}\) or \(R_{n}\) as indicated,express their limits as \(n \rightarrow \infty\) as definite integrals, identifying the correct intervals. $$R_{n}=\frac{1}{n} \sum_{i=1}^{n} \frac{i}{n}$$

In the following exercises, given \(L_{n}\) or \(R_{n}\) as indicated,express their limits as \(n \rightarrow \infty\) as definite integrals, identifying the correct intervals. $$L_{n}=\frac{2}{n} \sum_{i=1}^{n}\left(1+2 \frac{i-1}{n}\right)$$

In the following exercises, given \(L_{n}\) or \(R_{n}\) as indicated,express their limits as \(n \rightarrow \infty\) as definite integrals, identifying the correct intervals. $$R_{n}=\frac{3}{n} \sum_{i=1}^{n}\left(3+3 \frac{i}{n}\right)$$

In the following exercises, given \(L_{n}\) or \(R_{n}\) as indicated,express their limits as \(n \rightarrow \infty\) as definite integrals, identifying the correct intervals. $$L_{n}=\frac{2 \pi}{n} \sum_{i=1}^{n} 2 \pi \frac{i-1}{n} \cos \left(2 \pi \frac{i-1}{n}\right)$$

In the following exercises, given \(L_{n}\) or \(R_{n}\) as indicated,express their limits as \(n \rightarrow \infty\) as definite integrals, identifying the correct intervals. $$R_{n}=\frac{1}{n} \sum_{i=1}^{n}\left(1+\frac{i}{n}\right) \log \left(\left(1+\frac{i}{n}\right)^{2}\right)$$

In the following exercises, evaluate the integral using area formulas. \(\int_{0}^{3}(3-x) d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{2}^{3}(3-x) d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{-3}^{3}(3-|x|) d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{0}^{6}(3-|x-3|) d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{-3}^{3}(3-|x|) d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{1}^{5} \sqrt{4-(x-3)^{2}} d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{0}^{12} \sqrt{36-(x-6)^{2}} d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{-2}^{3}(3-|x|) d x\)

In the following exercises, use averages of values at the left (L) and right (R) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals. \(\\{(0,0),(2,1),(4,3),(5,0),(6,0),(8,3)\\} \quad\) over \([0,8]\)

In the following exercises, use averages of values at the left (L) and right (R) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals. \(\\{(0,2),(1,0),(3,5),(5,5),(6,2),(8,0)\\} \quad\) over \([0,8]\)

In the following exercises, use averages of values at the left (L) and right (R) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals. \(\\{(-4,-4),(-2,0),(0,-2),(3,3),(4,3)\\} \quad\) over \([-4,4]\)

In the following exercises, use averages of values at the left (L) and right (R) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals. \(\\{(-4,0),(-2,2),(0,0),(1,2),(3,2),(4,0)\\}\) over \([-4,4]\)

Suppose that \(\int_{0}^{4} f(x) d x=5\) and \(\int_{0}^{2} f(x) d x=-3,\) and \(\int_{0}^{4} g(x) d x=-1\) and \(\int_{0}^{2} g(x) d x=2 .\) In the following exercises, compute the integrals. \(\int_{0}^{4}(f(x)+g(x)) d x\)

Suppose that \(\int_{0}^{4} f(x) d x=5\) and \(\int_{0}^{2} f(x) d x=-3,\) and \(\int_{0}^{4} g(x) d x=-1\) and \(\int_{0}^{2} g(x) d x=2 .\) In the following exercises, compute the integrals. \(\int_{2}^{4}(f(x)+g(x)) d x\)

Suppose that \(\int_{0}^{4} f(x) d x=5\) and \(\int_{0}^{2} f(x) d x=-3,\) and \(\int_{0}^{4} g(x) d x=-1\) and \(\int_{0}^{2} g(x) d x=2 .\) In the following exercises, compute the integrals. \(\int_{0}^{2}(f(x)-g(x)) d x\)

Suppose that \(\int_{0}^{4} f(x) d x=5\) and \(\int_{0}^{2} f(x) d x=-3,\) and \(\int_{0}^{4} g(x) d x=-1\) and \(\int_{0}^{2} g(x) d x=2 .\) In the following exercises, compute the integrals. \(\int_{2}^{4}(f(x)-g(x)) d x\)

Suppose that \(\int_{0}^{4} f(x) d x=5\) and \(\int_{0}^{2} f(x) d x=-3,\) and \(\int_{0}^{4} g(x) d x=-1\) and \(\int_{0}^{2} g(x) d x=2 .\) In the following exercises, compute the integrals. \(\int_{0}^{2}(3 f(x)-4 g(x)) d x\)

Suppose that \(\int_{0}^{4} f(x) d x=5\) and \(\int_{0}^{2} f(x) d x=-3,\) and \(\int_{0}^{4} g(x) d x=-1\) and \(\int_{0}^{2} g(x) d x=2 .\) In the following exercises, compute the integrals. \(\int_{2}^{4}(4 f(x)-3 g(x)) d x\)

In the following exercises, use the identity \(\int_{-A}^{A} f(x) d x=\int_{-A}^{0} f(x) d x+\int_{0}^{A} f(x) d x\) to compute the integrals. \(\int_{-\pi}^{\pi} \frac{\sin t}{1+t^{2}} d t \quad(\text {Hint} : \sin (-t)=-\sin (t))\)

In the following exercises, use the identity \(\int_{-A}^{A} f(x) d x=\int_{-A}^{0} f(x) d x+\int_{0}^{A} f(x) d x\) to compute the integrals. \(\int_{-\sqrt{\pi}}^{\sqrt{\pi}} \frac{t}{1+\cos t} d t\)

In the following exercises, use the identity \(\int_{-A}^{A} f(x) d x=\int_{-A}^{0} f(x) d x+\int_{0}^{A} f(x) d x\) to compute the integrals. \(\int_{1}^{3}(2-x) d x(\text { Hint: Look at the graph of } f .)\)

In the following exercises, given that \(\int_{0}^{1} x d x=\frac{1}{2}, \int_{0}^{1} x^{2} d x=\frac{1}{3}, \quad\) and \(\quad \int_{0}^{1} x^{3} d x=\frac{1}{4}\) compute the integrals. \(\int_{0}^{1}\left(1+x+x^{2}+x^{3}\right) d x\)

In the following exercises, given that \(\int_{0}^{1} x d x=\frac{1}{2}, \int_{0}^{1} x^{2} d x=\frac{1}{3}, \quad\) and \(\quad \int_{0}^{1} x^{3} d x=\frac{1}{4}\) compute the integrals. \(\int_{0}^{1}\left(1-x+x^{2}-x^{3}\right) d x\)

In the following exercises, given that \(\int_{0}^{1} x d x=\frac{1}{2}, \int_{0}^{1} x^{2} d x=\frac{1}{3}, \quad\) and \(\quad \int_{0}^{1} x^{3} d x=\frac{1}{4}\) compute the integrals. \(\int_{0}^{1}(1-x)^{2} d x\)

In the following exercises, given that \(\int_{0}^{1} x d x=\frac{1}{2}, \int_{0}^{1} x^{2} d x=\frac{1}{3}, \quad\) and \(\quad \int_{0}^{1} x^{3} d x=\frac{1}{4}\) compute the integrals. \(\int_{0}^{1}(1-2 x)^{3} d x\)

In the following exercises, given that \(\int_{0}^{1} x d x=\frac{1}{2}, \int_{0}^{1} x^{2} d x=\frac{1}{3}, \quad\) and \(\quad \int_{0}^{1} x^{3} d x=\frac{1}{4}\) compute the integrals. \(\int_{0}^{1}\left(6 x-\frac{4}{3} x^{2}\right) d x\)

In the following exercises, given that \(\int_{0}^{1} x d x=\frac{1}{2}, \int_{0}^{1} x^{2} d x=\frac{1}{3}, \quad\) and \(\quad \int_{0}^{1} x^{3} d x=\frac{1}{4}\) compute the integrals. \(\int_{0}^{1}\left(7-5 x^{3}\right) d x\)

In the following exercises, use the comparison theorem. Show that \(\int_{0}^{3}\left(x^{2}-6 x+9\right) d x \geq 0\)

In the following exercises, use the comparison theorem. Show that \(\int_{-2}^{3}(x-3)(x+2) d x \leq 0\)

In the following exercises, use the comparison theorem. Show that \(\int_{0}^{1} \sqrt{1+x^{3}} d x \leq \int_{0}^{1} \sqrt{1+x^{2}} d x\)

In the following exercises, use the comparison theorem. Show that \(\int_{1}^{2} \sqrt{1+x} d x \leq \int_{1}^{2} \sqrt{1+x^{2}} d x\)

In the following exercises, use the comparison theorem. Show that \(\int_{0}^{\pi / 2} \sin t d t \geq \frac{\pi}{4} .\) (Hint: sin \(t \geq \frac{2 t}{\pi}\) over \(\left[0, \frac{\pi}{2}\right])\)

In the following exercises, use the comparison theorem. Show that \(\int_{-\pi / 4}^{\pi / 4} \cos t d t \geq \pi \sqrt{2} / 4\)

In the following exercises, find the average value \(f_{\text { ave }}\) of \(f\) between \(a\) and \(b,\) and find a point \(c,\) where \(f(c)=f_{\text { ave. }}\) \(f(x)=x^{2}, a=-1, b=1\)

In the following exercises, find the average value \(f_{\text { ave }}\) of \(f\) between \(a\) and \(b,\) and find a point \(c,\) where \(f(c)=f_{\text { ave. }}\) \(f(x)=x^{5}, a=-1, b=1\)

In the following exercises, find the average value \(f_{\text { ave }}\) of \(f\) between \(a\) and \(b,\) and find a point \(c,\) where \(f(c)=f_{\text { ave. }}\) \(f(x)=\sqrt{4-x^{2}}, a=0, b=2\)

In the following exercises, find the average value \(f_{\text { ave }}\) of \(f\) between \(a\) and \(b,\) and find a point \(c,\) where \(f(c)=f_{\text { ave. }}\) \(f(x)=(3-|x|), a=-3, b=3\)