미적분의 기본정리

Find the derivative.

$\frac{d}{dx}{\int }_{0}^{\cos x}{e}^{{t}^{2}}dt$

If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.

The average cost per gigabyte of RAM can be approximated by the function $C(t)=8,500,000{(0.65)}^{t},$ where $t$ is measured in years since 1980, and $C$ is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

Find the average cost of 1GB RAM for 2005 to 2010.

The velocity of a bullet from a rifle can be approximated by $v(t)=6400{t}^{2}-6505t+2686,$ where $t$ is seconds after the shot and $v$ is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: $0\le t\le 0.5.$ What is the total distance the bullet travels in 0.5 sec?

What is the average velocity of the bullet for the first half-second?

How are definite and indefinite integrals related?

What constant of integration is most commonly used when evaluating definite integrals?

T/F: If $f$ is a continuous function, then $\displaystyle F(x) = \int_a^x f(t)\ dt$ is also a continuous function.

The definite integral can be used to find “the area under a curve.” Give two other uses for definite integrals.

In Exercises 5–28, evaluate the definite integral.

$\displaystyle \int_1^3 (3x^2-2x+1)\ dx$

In Exercises 5–28, evaluate the definite integral.

$\displaystyle \int_0^4 (x-1)^2\ dx$

In Exercises 5–28, evaluate the definite integral.

$\displaystyle \int_{-1}^1 (x^3-x^5)\ dx$

In Exercises 5–28, evaluate the definite integral.

$\displaystyle \int_{\pi/2}^\pi \cos x\ dx$

In Exercises 5–28, evaluate the definite integral.

$\displaystyle \int_{0}^{\pi/4} \sec^2 x\ dx$

In Exercises 5–28, evaluate the definite integral.

$\displaystyle \int_{1}^{e} \frac1x \ dx$

In Exercises 5–28, evaluate the definite integral.

$\displaystyle \int_{-1}^{1} 5^x \ dx$

In Exercises 5–28, evaluate the definite integral.

$\displaystyle \int_{-2}^{-1} (4-2x^3) \ dx$

In Exercises 5–28, evaluate the definite integral.

$\displaystyle \int_{0}^{\pi} (2\cos x - 2\sin x) \ dx$

In Exercises 5–28, evaluate the definite integral.

$\displaystyle \int_{1}^{3} e^x \ dx$