Applications of Integration Practice Questions
AP (Advanced Placement) · AP Calculus AB · 150 free MCQs with instant results and detailed explanations.
150
Total
45
Easy
83
Medium
22
Hard
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Sample Questions from Applications of Integration
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
If a function G(t) represents the amount of water flowing into a tank over time, which of the following represents the total amount of water in the tank after 5 hours, assuming it starts empty?
A.
โซ(0 to 5) G(t) dt
B.
G(5)
C.
G(0) + G(5)
D.
โซ(5 to 0) G(t) dt
Show Answer & Explanation
Correct Answer: A
The total amount of water in the tank after 5 hours is represented by the definite integral of G(t) from 0 to 5, which calculates the accumulated inflow of water over that period.
Q2
Easy
A rectangle is inscribed under the curve y = x^2 from x = 0 to x = 3. What is the maximum area of the rectangle?
A.
9
B.
12
C.
6
D.
3
Show Answer & Explanation
Correct Answer: A
The area of the rectangle can be expressed as A = x * f(x) = x * x^2 = x^3. To maximize the area, we take the derivative, set it to zero, and find x = 3, resulting in maximum area A = 9.
Q3
Easy
What is the area under the curve of the function f(x) = 2x from x = 1 to x = 3?
A.
8
B.
6
C.
4
D.
2
Show Answer & Explanation
Correct Answer: A
The area under the curve can be computed using the definite integral from 1 to 3. โซ(2x)dx from 1 to 3 = [x^2] from 1 to 3 = 3^2 - 1^2 = 9 - 1 = 8.
Q4
Medium
What is the area of the region bounded by the curve y = x^2 and the x-axis from x = 0 to x = 3?
A.
9
B.
27
C.
4.5
D.
12
Show Answer & Explanation
Correct Answer: A
To find the area under the curve y = x^2 from 0 to 3, we calculate the definite integral โซ from 0 to 3 of x^2 dx, which evaluates to (1/3)x^3 from 0 to 3 = (1/3)(27) - (1/3)(0) = 9.
Q5
Medium
The volume of the solid formed by rotating the region between y = x^2 and y = x around the x-axis from x = 0 to x = 1 is?
A.
ฯ/5
B.
ฯ/3
C.
ฯ/4
D.
ฯ/6
Show Answer & Explanation
Correct Answer: A
The volume can be found using the disk method: V = ฯโซ from 0 to 1 ((x^2)^2 - (x)^2) dx = ฯโซ from 0 to 1 (x^4 - x^2) dx = ฯ[(1/5)(1^5) - (1/3)(1^3)] = ฯ(1/5 - 1/3) = ฯ(3-5)/15 = ฯ/15.
Q6
Medium
What is the total area between the curve y = x^2 and the x-axis from x = 1 to x = 3?
A.
8/3
B.
10/3
C.
7/2
D.
16/3
Show Answer & Explanation
Correct Answer: B
The area under the curve from x = 1 to x = 3 is found by integrating y = x^2 from 1 to 3. The integral evaluates to (1/3)(3^3) - (1/3)(1^3) = 10/3.
Q7
Medium
A solid is generated by rotating the region bounded by y = x^2 and y = 4 about the y-axis. What is the volume of this solid?
A.
32ฯ/15
B.
16ฯ/3
C.
20ฯ/3
D.
8ฯ/3
Show Answer & Explanation
Correct Answer: B
Using the method of cylindrical shells, the volume is determined by the integral 2ฯโซ(x)(4-x^2)dx from 0 to 2, which evaluates to 16ฯ/3.
Q8
Hard
A tank is being filled with water. The rate of water flow into the tank is given by the function R(t) = 5t^2 - 4t + 3 liters per hour, where t is the time in hours. What is the total amount of water in the tank after 3 hours?
A.
48 liters
B.
54 liters
C.
36 liters
D.
60 liters
Show Answer & Explanation
Correct Answer: B
To find the total amount of water, we need to integrate the rate function R(t) from 0 to 3. The integral of R(t) gives us the total volume, which calculates to 54 liters after evaluation.
Q9
Hard
The area between the curve y = x^3 and the x-axis from x = 1 to x = 2 is to be calculated. What is the value of this area?
A.
1.25
B.
2.25
C.
3.75
D.
2.00
Show Answer & Explanation
Correct Answer: C
The area under the curve can be found by integrating the function y = x^3 from 1 to 2. The definite integral evaluates to 3.75, representing the area between the curve and the x-axis.
Q10
Hard
A region in the first quadrant is bounded by the curve y = 4 - x^2 and the x-axis. What is the area of this region?
A.
8/3
B.
16/3
C.
32/3
D.
4
Show Answer & Explanation
Correct Answer: B
The area is calculated using the integral from 0 to 2 of (4 - x^2) dx. This yields the area of the specified region as 16/3.
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Applications of Integration โ AP (Advanced Placement) AP Calculus AB Practice Questions Online
This page contains 150 practice MCQs for the chapter Applications of Integration in AP (Advanced Placement) AP Calculus AB. The questions are organized by difficulty โ 45 easy, 83 medium, 22 hard โ so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.