미적분의 기본정리

In the following exercises, use a calculator to estimate the area under the curve by computing T₁₀, the average of the left- and right-endpoint Riemann sums using $N=10$ rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area.

[T] $y={x}^{3}+6{x}^{2}+x-5$ over $[-4,2]$

In the following exercises, use a calculator to estimate the area under the curve by computing T₁₀, the average of the left- and right-endpoint Riemann sums using $N=10$ rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area.

[T] $y=\sqrt{{x}^{3}}$ over $[0,6]$

In the following exercises, use a calculator to estimate the area under the curve by computing T₁₀, the average of the left- and right-endpoint Riemann sums using $N=10$ rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area.

[T] $y=\sqrt{x}+{x}^{2}$ over $[1,9]$

In the following exercises, use a calculator to estimate the area under the curve by computing T₁₀, the average of the left- and right-endpoint Riemann sums using $N=10$ rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area.

[T] $\int (\cos x-\sin x)dx$ over $[0,\pi ]$

In the following exercises, use a calculator to estimate the area under the curve by computing T₁₀, the average of the left- and right-endpoint Riemann sums using $N=10$ rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area.

[T] $\int \frac{4}{{x}^{2}}dx$ over $[1,4]$

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

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

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

${\int }_{-2}^{3}({x}^{2}+3x-5)dx$

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

${\int }_{-2}^{3}(t+2)(t-3)dt$

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

${\int }_{2}^{3}({t}^{2}-9)(4-{t}^{2})dt$

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

${\int }_{1}^{2}{x}^{9}dx$

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

${\int }_{0}^{1}{x}^{99}dx$

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

${\int }_{4}^{8}(4{t}^{5/2}-3{t}^{3/2})dt$

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

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

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

${\int }_{1}^{2}\frac{2}{{x}^{3}}dx$

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

${\int }_{1}^{4}\frac{1}{2\sqrt{x}}dx$

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.

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