적분 전체 문제

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2.

${\int }_{1}^{4}\frac{2-\sqrt{t}}{{t}^{2}}dt$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2.

${\int }_{1}^{16}\frac{dt}{{t}^{1/4}}$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2.

${\int }_{0}^{2\pi }\cos \theta d\theta$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2.

${\int }_{0}^{\pi /4}{\sec }^{2}\theta d\theta$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2.

${\int }_{0}^{\pi /4}\sec \theta \tan \theta d\theta$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2.

${\int }_{\pi /3}^{\pi /4}\csc \theta \cot \theta d\theta$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2.

${\int }_{\pi /4}^{\pi /2}{\csc }^{2}\theta d\theta$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2.

${\int }_{1}^{2}\left(\frac{1}{{t}^{2}}-\frac{1}{{t}^{3}}\right)dt$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2.

${\int }_{-2}^{-1}\left(\frac{1}{{t}^{2}}-\frac{1}{{t}^{3}}\right)dt$

In the following exercises, use the evaluation theorem to express the integral as a function $F(x).$

${\int }_{a}^{x}{t}^{2}dt$

In the following exercises, use the evaluation theorem to express the integral as a function $F(x).$

${\int }_{1}^{x}{e}^{t}dt$

In the following exercises, use the evaluation theorem to express the integral as a function $F(x).$

${\int }_{0}^{x}\cos tdt$

In the following exercises, use the evaluation theorem to express the integral as a function $F(x).$

${\int }_{-x}^{x}\sin tdt$

In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2.

${\int }_{-2}^{3}|x|dx$

In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2.

${\int }_{-2}^{4}|{t}^{2}-2t-3|dt$

In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2.

${\int }_{0}^{\pi }|\cos t|dt$

In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2.

${\int }_{-\pi /2}^{\pi /2}|\sin t|dt$

Suppose the rate of gasoline consumption over the course of a year in the United States can be modeled by a sinusoidal function of the form $\left(11.21-\cos \left(\frac{\pi t}{6}\right)\right)\,\times \,{10}^{9}$ gal/mo.

a. What is the average monthly consumption, and for which values of t is the rate at time t equal to the average rate? b. What is the number of gallons of gasoline consumed in the United States in a year? c. Write an integral that expresses the average monthly U.S. gas consumption during the part of the year between the beginning of April $(t=3)$ and the end of September $(t=9).$

Explain why, if f is continuous over $[a,b],$ there is at least one point $c\in [a,b]$ such that $f(c)=\frac{1}{b-a}{\int }_{a}^{b}f(t)dt.$

Explain why, if f is continuous over $[a,b]$ and is not equal to a constant, there is at least one point $M\in [a,b]$ such that $f(M)>\frac{1}{b-a}{\int }_{a}^{b}f(t)dt$ and at least one point $m\in [a,b]$ such that $f(m)<\frac{1}{b-a}{\int }_{a}^{b}f(t)dt.$