적분 전체 문제

For the following exercises, solve the initial value problem.

${f}^{'}(x)=\frac{2}{{x}^{2}}-\frac{{x}^{2}}{2},f(1)=0$

A car is being driven at a rate of $40$ mph when the brakes are applied. The car decelerates at a constant rate of $10$ ft/sec². How long before the car stops?

In the preceding problem, calculate how far the car travels in the time it takes to stop.

A car company wants to ensure its newest model can stop in $8$ sec when traveling at $75$ mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.

A car company wants to ensure its newest model can stop in less than $450$ ft when traveling at $60$ mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.

For the following exercises, find the antiderivative of the function, assuming $F(0)=0.$

[T] $f(x)={x}^{2}+2$

For the following exercises, find the antiderivative of the function, assuming $F(0)=0.$

[T] $f(x)=4x-\sqrt{x}$

For the following exercises, find the antiderivative of the function, assuming $F(0)=0.$

[T] $f(x)=\sin x+2x$

For the following exercises, find the antiderivative of the function, assuming $F(0)=0.$

[T] $f(x)={e}^{x}$

For the following exercises, find the antiderivative of the function, assuming $F(0)=0.$

[T] $f(x)={e}^{-2x}+3{x}^{2}$

If $f(x)$ is the antiderivative of $v(x),$ then $2f(x)$ is the antiderivative of $2v(x).$

If $f(x)$ is the antiderivative of $v(x),$ then $f(2x)$ is the antiderivative of $v(2x).$

If $f(x)$ is the antiderivative of $v(x),$ then $f(x)+1$ is the antiderivative of $v(x)+1.$

If $f(x)$ is the antiderivative of $v(x),$ then ${(f(x))}^{2}$ is the antiderivative of ${(v(x))}^{2}.$

In the following exercises, express the limits as integrals.

$\lim_{n\to \infty }\sum_{i=1}^{n}\left({x}_{i}^{*}\right)\Delta x$ over $[1,3]$

In the following exercises, express the limits as integrals.

$\lim_{n\to \infty }\sum_{i=1}^{n}\left(5{\left({x}_{i}^{*}\right)}^{2}-3{\left({x}_{i}^{*}\right)}^{3}\right)\Delta x$ over $[0,2]$

In the following exercises, express the limits as integrals.

$\lim_{n\to \infty }\sum_{i=1}^{n}{\sin }^{2}\left(2\pi {x}_{i}^{*}\right)\Delta x$ over $[0,1]$

In the following exercises, express the limits as integrals.

$\lim_{n\to \infty }\sum_{i=1}^{n}{\cos }^{2}\left(2\pi {x}_{i}^{*}\right)\Delta x$ over $[0,1]$

In the following exercises, given L_n or R_n as indicated, express their limits as $n\to \infty$ as definite integrals, identifying the correct intervals.

${L}_{n}=\frac{1}{n}\sum_{i=1}^{n}\frac{i-1}{n}$

In the following exercises, given L_n or R_n as indicated, express their limits as $n\to \infty$ as definite integrals, identifying the correct intervals.

${L}_{n}=\frac{2}{n}\sum_{i=1}^{n}\left(1+2\frac{i-1}{n}\right)$