AB Calculus Applications of Integrals

The population of a town grows at a rate of r(t) people per year (where t is time in years).  What does  $\int_2^4r\left(t\right)dt$  represent?

• The time it took for the town to grow from a population of 2 people to a population of 4 people.

• The number of people in the town on the fourth year.

• The average rate at which the population grew between the second and the fourth year.

• The change in the number of people between the second and the fourth year.

Which expression represents the sum of the areas of regions R and S.

• $\int_0^{4.5}\left(f\left(x\right)-g\left(x\right)\right)dx$

• $\int_0^{1.3733}\left(f\left(x\right)-g\left(x\right)\right)dx+\int_{1.3733}^{4.5}\left(g\left(x\right)-f\left(x\right)\right)dx$

• $\int_0^{1.3733}\left(g\left(x\right)-f\left(x\right)\right)dx+\int_{1.3733}^{4.5}\left(f\left(x\right)-g\left(x\right)\right)dx$

• $2\cdot\int_0^{1.3733}\left(f\left(x\right)-g\left(x\right)\right)dx$

What is the area of the region between the graphs of  $f\left(x\right)=8x+6$   and  $g\left(x\right)=x-x^2$  from x= -6 and x = -1?

• 485/6

• 1195/6

• 125/6

• 643/6

Which one of the definite integrals below gives the volume of the solid?

• $\int_{-1}^3\left(-x^2+6x-5\right)^2dx$

• $\int_{-1}^5\left(-x^2+6x-5\right)^2dx$

• $\int_1^3\left(-x^2+6x-5\right)^2dx$

• $\int_1^5\left(-x^2+6x-5\right)^2dx$

Which one of the definite integrals below gives the volume of the solid?

• $\pi\int_0^3e^{2y}dy$

• $\pi\int_0^3\left[\ln\left(y\right)\right]^2dy$

• $\pi\int_0^{\ln3}e^{2y}dy$

• $\pi\int_0^{e^3}\left[\ln\left(y\right)\right]^2dy$

Which one of the definite integrals gives the volume of the solid?

• $\pi\int_1^4\left[e^y+4\right]^2dy$

• $\pi\int_1^4\left[\ln\left(y\right)-4\right]dy$

• $\pi\int_1^4\left[\ln\left(y\right)+4\right]^2dy$

• $\pi\int_1^4\left[e^y-4\right]^2dy$

 The temperature in a room changes at a rate of  $r\left(t\right)=\ln\left(t+1\right)\sin\left(t\right)$   degrees Celsius per hour (where t is the time in hours).  At t = 1 hour, the temperature is 18 degrees Celsius.
What is the room's temperature at time t = 3 hours?

• 1.571 degrees C

• 19.571 degrees C

• 19.323 degrees C

• 1.323 degrees C

What is the volume of the solid generated when R is rotated about the x-axis?

• $8\pi$

• $6\pi$

• $\frac{8}{3}\pi$

• $\frac{8}{3}$

The area of the region enclosed by the graphs of  $y=e^{x^2}-2$   and  $y=\sqrt{4-x^2}$   is

• 2.525

• 4.049

• 5.050

• 6.289

The region in the first quadrant enclosed by the graphs of  $y=x$  and  $y=2\sin x$  is revolved about the x-axis.  The volume of the solid generated is 

• 1.433

• 1.895

• 5.811

• 6.678