적분 전체 문제

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

${\int }_{2}^{4}(f(x)-g(x))dx$

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

${\int }_{0}^{2}(3f(x)-4g(x))dx$

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

${\int }_{2}^{4}(4f(x)-3g(x))dx$

In the following exercises, use the identity ${\int }_{-A}^{A}f(x)dx={\int }_{-A}^{0}f(x)dx+{\int }_{0}^{A}f(x)dx$ to compute the integrals.

${\int }_{-\pi }^{\pi }\frac{\sin t}{1+{t}^{2}}dt$ $(Hint:\,\sin (-t)=-\sin (t))$

In the following exercises, find the net signed area between $f(x)$ and the x-axis.

${\int }_{1}^{3}(2-x)dx$ (Hint: Look at the graph of f.)

In the following exercises, find the net signed area between $f(x)$ and the x-axis.

${\int }_{2}^{4}{(x-3)}^{3}dx$ (Hint: Look at the graph of f.)

In the following exercises, given that ${\int }_{0}^{1}xdx=\frac{1}{2},{\int }_{0}^{1}{x}^{2}dx=\frac{1}{3},$ and ${\int }_{0}^{1}{x}^{3}dx=\frac{1}{4},$ compute the integrals.

${\int }_{0}^{1}(1+x+{x}^{2}+{x}^{3})dx$

In the following exercises, given that ${\int }_{0}^{1}xdx=\frac{1}{2},{\int }_{0}^{1}{x}^{2}dx=\frac{1}{3},$ and ${\int }_{0}^{1}{x}^{3}dx=\frac{1}{4},$ compute the integrals.

${\int }_{0}^{1}(1-x+{x}^{2}-{x}^{3})dx$

In the following exercises, given that ${\int }_{0}^{1}xdx=\frac{1}{2},{\int }_{0}^{1}{x}^{2}dx=\frac{1}{3},$ and ${\int }_{0}^{1}{x}^{3}dx=\frac{1}{4},$ compute the integrals.

${\int }_{0}^{1}{(1-x)}^{2}dx$

In the following exercises, given that ${\int }_{0}^{1}xdx=\frac{1}{2},{\int }_{0}^{1}{x}^{2}dx=\frac{1}{3},$ and ${\int }_{0}^{1}{x}^{3}dx=\frac{1}{4},$ compute the integrals.

${\int }_{0}^{1}\left(6x-\frac{4}{3}{x}^{2}\right)dx$

In the following exercises, given that ${\int }_{0}^{1}xdx=\frac{1}{2},{\int }_{0}^{1}{x}^{2}dx=\frac{1}{3},$ and ${\int }_{0}^{1}{x}^{3}dx=\frac{1}{4},$ compute the integrals.

${\int }_{0}^{1}(7-5{x}^{3})dx$

In the following exercises, use the comparison theorem.

Show that ${\int }_{0}^{3}({x}^{2}-6x+9)dx\ge 0.$

In the following exercises, use the comparison theorem.

Show that ${\int }_{-2}^{3}(x-3)(x+2)dx\le 0.$

In the following exercises, use the comparison theorem.

Show that ${\int }_{0}^{1}\sqrt{1+{x}^{3}}dx\le {\int }_{0}^{1}\sqrt{1+{x}^{2}}dx.$

In the following exercises, use the comparison theorem.

Show that ${\int }_{1}^{2}\sqrt{1+x}dx\le {\int }_{1}^{2}\sqrt{1+{x}^{2}}dx.$

In the following exercises, use the comparison theorem.

Show that ${\int }_{0}^{\pi /2}\sin tdt\ge \frac{\pi }{4}.$ $(Hint:\,\sin t\ge \frac{2t}{\pi }$ over $\left[0,\frac{\pi }{2}\right])$

In the following exercises, use the comparison theorem.

Show that ${\int }_{-\pi /4}^{\pi /4}\cos tdt\ge \pi \sqrt{2}/4.$

In the following exercises, find the average value f_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_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_ave of f between a and b, and find a point c, where $f(c)={f}_{\text{ave}}.$

$f(x)=\sqrt{4-{x}^{2}},a=0,b=2$