Chapter 5

Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a system, evaluate the 10 -subinterval Riemann sums using left end point, right end point, and midpoint evaluations. $$ \int_{0}^{1} \tan x d x $$

Let \(x_{1}, x_{2}, \ldots, x_{n}\) be any real numbers. Find the value of \(c\) that minimizes \(\sum_{i=1}^{n}\left(x_{i}-c\right)^{2}\).

Use symmetry to help you evaluate the given integral. $$ \int_{-\pi / 2}^{\pi / 2} \frac{\sin x}{1+\cos x} d x $$

use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{-1}^{3} \frac{1}{(t+2)^{2}} d t $$

Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a system, evaluate the 10 -subinterval Riemann sums using left end point, right end point, and midpoint evaluations. $$ \int_{0}^{1} \cos x d x $$

In the song The Twelve Days of Christmas, my true love gave me 1 gift on the first day, \(1+2\) gifts on the second day, \(1+2+3\) gifts on the third day, and so on for 12 days. (a) Find the total number of gifts given in 12 days. (b) Find a simple formula for \(T_{n}\), the total number of gifts given during a Christmas of \(n\) days.

Use symmetry to help you evaluate the given integral. $$ \int_{-\sqrt[3]{\pi}}^{\sqrt[3]{\pi}} x^{2} \cos \left(x^{3}\right) d x $$

use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{2}^{10} \frac{1}{y+4} d y $$

Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a system, evaluate the 10 -subinterval Riemann sums using left end point, right end point, and midpoint evaluations. $$ \int_{1}^{3}(1 / x) d x $$

A grocer stacks oranges in a pyramidlike pile. If the bottom layer is rectangular with 10 rows of 16 oranges and the top layer has a single row of oranges, how many oranges are in the stack?

Use symmetry to help you evaluate the given integral. $$ \int_{-\pi}^{\pi}(\sin x+\cos x)^{2} d x $$

Let \(F(x)=\int_{0}^{x}\left(t^{4}+1\right) d t\). (a) Find \(F(0)\). (b) Let \(y=F(x)\). Apply the First Fundamental Theorem of Calculus to obtain \(d y / d x=F^{\prime}(x)=x^{4}+1 .\) Solve the differential equation \(d y / d x=x^{4}+1\). (c) Find the solution to this differential equation that satisfies \(y=F(0)\) when \(x=0\). (d) Show that \(\int_{0}^{1}\left(x^{4}+1\right) d x=\frac{6}{5}\).

use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{5}^{8} \sqrt{3 x+1} d x $$

Prove that the function \(f\) defined by $$f(x)=\left\\{\begin{array}{ll} 1 & \text { if } x \text { is rational } \\ 0 & \text { if } x \text { is irrational } \end{array}\right.$$ is not integrable on \([0,1] .\) Hint \(:\) Show that no matter how small the norm of the partition, \(\|P\|\), the Riemann sum can be made to have value either 0 or 1

Let \(G(x)=\int_{0}^{x} \sin t d t\). (a) Find \(G(0)\) and \(G(2 \pi)\). (b) Let \(y=G(x) .\) Apply the First Fundamental Theorem of Calculus to obtain \(d y / d x=G^{\prime}(x)=\sin x .\) Solve the differential equation \(d y / d x=\sin x\) (c) Find the solution to this differential equation that satisfies \(y=G(0)\) when \(x=0\) (d) Show that \(\int_{0}^{\pi} \sin x d x=2\). (e) Find all relative extrema and inflection points of \(G\) on the interval \([0,4 \pi]\) (f) Plot a graph of \(y=G(x)\) over the interval \([0,4 \pi]\).

Use symmetry to help you evaluate the given integral. $$ \int_{-\pi / 2}^{\pi / 2} z \sin ^{2}\left(z^{3}\right) \cos \left(z^{3}\right) d z $$

use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{1}^{7} \frac{1}{\sqrt{2 x+2}} d x $$

use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{-3}^{3} \sqrt{7+2 t^{2}}(8 t) d t $$

Find a nice formula for the sum $$ \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)} $$

Use symmetry to help you evaluate the given integral. $$ \int_{-100}^{100}\left(v+\sin v+v \cos v+\sin ^{3} v\right)^{5} d v $$

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{1}^{3} \frac{x^{2}+1}{\sqrt{x^{3}+3 x}} d x $$

Use symmetry to help you evaluate the given integral. $$ \int_{-1}^{1} x e^{-4 x^{2}} d x $$

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{\pi / 2} \cos ^{2} x \sin x d x $$

Use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{2}^{4}(x+6)^{5} d x $$

Use symmetry to help you evaluate the given integral. $$ \int_{-\pi / 4}^{\pi / 4}\left(|x| \sin ^{5} x+|x|^{2} \tan x\right) d x $$

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{\pi / 2} \sin ^{2} 3 x \cos 3 x d x $$

How does \(\int_{-b}^{-a} f(x) d x\) compare with \(\int_{a}^{b} f(x) d x\) when \(f\) is an even function? An odd function?

Use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{1}^{5}\left(3+\frac{2}{x}\right) d x $$

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1} x e^{x^{2}} d x $$

Prove (by a substitution) that $$ \int_{a}^{b} f(-x) d x=\int_{-b}^{-a} f(x) d x $$

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{1}^{4} \frac{(\sqrt{x}-1)^{3}}{\sqrt{x}} d x $$

Use periodicity to calculate \(\int_{0}^{4 \pi}|\cos x| d x\).

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{\pi / 6} \sin ^{3} \theta \cos \theta d \theta $$

Calculate \(\int_{0}^{4 \pi}|\sin 2 x| d x\).

Use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{0.2}^{0.4}\left(0.002+0.0001 \cos ^{2} x\right) d x $$

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{\pi / 6} \frac{\sin \theta}{\cos ^{3} \theta} d \theta $$

Sketch the graph of the given function over the interval \([a, b] ;\) then divide \([a, b]\) into \(n\) equal subintervals. Finally, calculate the area of the corresponding circumscribed polygon. $$ f(x)=x+1 ; a=-1, b=2, n=3 $$

If \(f\) is periodic with period \(p\), then $$ \int_{a}^{a+p} f(x) d x=\int_{0}^{p} f(x) d x $$ Convince yourself that this is true by drawing a picture and then use the result to calculate \(\int_{1}^{1+\pi}|\sin x| d x\).

Find \(\lim _{x \rightarrow 0} \frac{1}{x} \int_{0}^{x} \frac{1+t}{2+t} d t\).

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1} \cos (3 x-3) d x $$

Sketch the graph of the given function over the interval \([a, b] ;\) then divide \([a, b]\) into \(n\) equal subintervals. Finally, calculate the area of the corresponding circumscribed polygon. $$ f(x)=3 x-1 ; a=1, b=3, n=4 $$

Find \(\lim _{x \rightarrow 1} \frac{1}{x-1} \int_{1}^{x} \frac{1+t}{2+t} d t\).

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1 / 2} \sin (2 \pi x) d x $$

Sketch the graph of the given function over the interval \([a, b] ;\) then divide \([a, b]\) into \(n\) equal subintervals. Finally, calculate the area of the corresponding circumscribed polygon. $$ f(x)=x^{2}-1 ; a=2, b=3, n=6 $$

Calculate \(\int_{1}^{1+\pi}|\cos x| d x\).

Find \(f(x)\) if \(\int_{1}^{x} f(t) d t=2 x-2\).

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1} x \sin \left(\pi x^{2}\right) d x $$

Sketch the graph of the given function over the interval \([a, b] ;\) then divide \([a, b]\) into \(n\) equal subintervals. Finally, calculate the area of the corresponding circumscribed polygon. $$ f(x)=3 x^{2}+x+1 ; a=-1, b=1, n=10 $$

Prove or disprove that the integral of the average value equals the integral of the function on the interval: \(\int_{a}^{b} \bar{f} d x=\) \(\int_{a}^{b} f(x) d x\), where \(\bar{f}\) is the average value of the function \(f\) over the interval \([a, b]\).

Find \(f(x)\) if \(\int_{0}^{x} f(t) d t=x^{2}\).