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AP Calculus BC : Application of Integrals

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Example Questions

Example Question #1 : Length Of Curve, Distance Traveled, Accumulated Change, Motion Of Curve

Find the total distance traveled by a particle along the curve y=x^2 from x=0 to x=2 .

Possible Answers:

$\int_0^2\sqrt{1+2x^2}dx$

$\int_0^2\sqrt{1+x^2}dx$

$\int_0^2\sqrt{1+4x^2}dx$

$\int_0^2\sqrt{1+2x}dx$

Correct answer:

$\int_0^2\sqrt{1+4x^2}dx$

Explanation:

To find the required distance, we can use the arc length expression given by $\int_{x_0}^{x_1}\sqrt{1+f'(x)^2}dx$ .

Taking the derivative of our function, we have y'=f'(x)=2x . Plugging in our values for our integral bounds, we have

$\int_0^2\sqrt{1+(2x)^2}dx = \int_0^2 \sqrt{1+4x^2}dx$ .

As with most arc length integrals, this integral is too difficult (if not, outright impossible) to evaluate explicitly by hand. So we will just leave it this form, or evaluate it with some computer software.

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Example Question #1 : Modeling By Solving Separable Differential Equations

Solve the separable differential equation

$\frac{\mathrm{d} y}{\mathrm{d} x}= yx^2$

with the condition y(0)=4 .

Possible Answers:

$y=4e^{\frac{x^3}{3}}$

$y=Ce^{\frac{x^2}{2}}$

$y=Ce^{\frac{x^3}{3}}$

$y=Ce^{\frac{64}{3}}$

$y=4e^{\frac{x^2}{3}}$

Correct answer:

$y=4e^{\frac{x^3}{3}}$

Explanation:

To solve the separable differential equation, we must separate x and y, dx and dy respectively to opposite sides:

$\frac{dy}{y}=x^2dx$

Integrating both sides, we get

$\ln\left | y\right |=\frac{ x^3}{3}+C$

The rules of integration used were

$\int \frac{dx}{x}=\ln\left | x\right |+C$ , $\int x^ndx= \frac{x^{n+1}}{n+1}+C$

The constants of integration merged into one.

Now, we exponentiate both sides of the equation to solve for y, and use the properties of exponents to simplify:

$y=e^{\frac{x^3}{3}}+e^C=e^{{\frac{x^3}{3}}\cdot C}=Ce^{\frac{x^3}{3}}$

To solve for C, we use our given condition:

4=Ce^0

4=C

Our final answer is

$y=4e^{\frac{x^3}{3}}$

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AP Calculus BC : Application of Integrals

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #1 : Length Of Curve, Distance Traveled, Accumulated Change, Motion Of Curve

Example Question #1 : Modeling By Solving Separable Differential Equations

All AP Calculus BC Resources

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