Applications of Integrals: Interactive Learning Quizzes for Advanced Mathematics

Question 1

A solid is generated by revolving the region bounded by the curve y = √x, the x-axis, and the line x = 4 about the x-axis. What is the volume of this solid?

Accepted Answer

8π

Answer

32π

Answer

16π

Answer

4π

Question 2

A spring has a natural length of 10 cm. A force of 20 N is required to stretch the spring to a length of 12 cm. How much work is done in stretching the spring from its natural length to a length of 15 cm?

Accepted Answer

15 J

Answer

20 J

Answer

25 J

Answer

10 J

Question 3

A thin plate is bounded by the curves y = x^2 and y = x. The density of the plate at any point (x, y) is given by δ(x, y) = xy. What is the mass of the plate?

Accepted Answer

1/12

Answer

1/6

Answer

1/2

Answer

1/3

Question 4

Find the length of the curve y = (x^3/3) + (1/4x) from x = 1 to x = 2.

Accepted Answer

17/12

Answer

3/2

Answer

5/4

Answer

7/3

Question 5

A region in the first quadrant is bounded by the curves y = x^2, y = 4, and the y-axis. What is the x-coordinate of the centroid of this region?

Accepted Answer

8/5

Answer

2/3

Answer

3/2

Answer

4/3

Question 6

The region bounded by the curves y = sin(x), y = 0, and x = π is revolved about the x-axis. What is the surface area of the resulting solid?

Accepted Answer

2π

Answer

π

Answer

8π

Answer

4π

Question 7

A tank in the shape of a right circular cone is filled with water to a depth of 1 meter. The tank has a radius of 2 meters at the top and a height of 3 meters. How much work is done in pumping all the water out of the tank over the top?

Accepted Answer

19600π J

Answer

78400π J

Answer

9800π J

Answer

39200π J

Question 8

Find the volume of the solid generated by rotating the region bounded by the curves y = e^x, y = 0, x = 0, and x = 1 about the y-axis.

Accepted Answer

π(e^2 - 1)

Answer

π(e^2 + 1)

Answer

π(e + 1)

Answer

π(e - 1)

Question 9

A thin wire is bent into the shape of a semicircle of radius 2. The density of the wire at a point P is proportional to the distance from P to the diameter of the semicircle. Find the mass of the wire.

Accepted Answer

8k

Answer

16k

Answer

4k

Question 10

A lamina occupies the region bounded by the curves y = x^2 and y = 2x. The density at any point (x, y) is given by δ(x, y) = x. Find the moment of inertia of the lamina about the y-axis.

Accepted Answer

16/15

Answer

2/15

Answer

4/15

Answer

8/15