Integration and Accumulation Practice Questions
AP (Advanced Placement) · AP Calculus AB · 146 free MCQs with instant results and detailed explanations.
146
Total
34
Easy
76
Medium
36
Hard
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Sample Questions from Integration and Accumulation
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Q1
Easy
What is the integral of the function f(x) = 3x^2?
A.
x^3 + C
B.
x^3 + 3C
C.
6x + C
D.
x^2 + 3C
Show Answer & Explanation
Correct Answer: A
The integral of 3x^2 is found using the power rule, which states that โซx^n dx = (x^(n+1))/(n+1) + C. Here, n=2, so โซ3x^2 dx = 3*(x^(2+1))/(2+1) + C = x^3 + C.
Q2
Easy
Which of the following functions represents the accumulated area function A(t) for f(x) = 2x, starting from x = 0?
A.
x^2
B.
x^2 + C
C.
x^2/2
D.
2x^2
Show Answer & Explanation
Correct Answer: A
The accumulated area function A(t) is found by integrating f(x) = 2x from 0 to t: A(t) = โซ(from 0 to t) 2x dx = [x^2] (from 0 to t) = t^2 - 0^2 = t^2.
Q3
Easy
What is the integral of the function f(x) = 3xยฒ with respect to x?
A.
xยณ + C
B.
3xยณ + C
C.
xยฒ + C
D.
6x + C
Show Answer & Explanation
Correct Answer: A
The integral of 3xยฒ is calculated using the power rule, which states that โซx^n dx = (x^(n+1))/(n+1) + C. Here, n is 2, so the integral becomes (3x^(2+1))/(2+1) + C = xยณ + C.
Q4
Medium
Evaluate the integral โซ(sin(x^2)) dx from 0 to โฯ.
A.
โ2
B.
1
C.
1/2
D.
โฯ/2
Show Answer & Explanation
Correct Answer: D
The integral of sin(x^2) does not have a simple antiderivative, but using the substitution u = x^2, the limits change from 0 to ฯ, giving โฯ/2 as the value.
Q5
Medium
Given the function f(x) = x^3 - 6x^2 + 9x, what is the definite integral of f from 1 to 3?
A.
4
B.
8
C.
6
D.
10
Show Answer & Explanation
Correct Answer: B
The definite integral from 1 to 3 calculates the area under the curve of f(x). Evaluating the integral gives us 8.
Q6
Medium
If F(x) = โซ(2x + 1)dx from 0 to x, what is F(2)?
A.
8
B.
6
C.
4
D.
5
Show Answer & Explanation
Correct Answer: B
The integral of (2x + 1) from 0 to x gives F(x) = x^2 + x. Thus, F(2) = 2^2 + 2 = 6.
Q7
Medium
The rate of water flowing into a tank is given by the function R(t) = 5t^2 + 3, where t is in hours. What is the total amount of water in the tank after 2 hours?
A.
22
B.
30
C.
18
D.
26
Show Answer & Explanation
Correct Answer: D
To find the total amount of water after 2 hours, we integrate R(t) from 0 to 2. The result is 26.
Q8
Hard
If F(x) is an antiderivative of f(x) on the interval [a, b], which of the following statements is correct regarding the integral of f(x) from a to b?
A.
The integral is equal to F(b) - F(a).
B.
The integral is equal to F(a) - F(b).
C.
The integral is equal to F'(b) - F'(a).
D.
The integral can be found by evaluating f(b) - f(a).
Show Answer & Explanation
Correct Answer: A
According to the Fundamental Theorem of Calculus, if F is an antiderivative of f on [a, b], then the definite integral from a to b of f(x) dx is F(b) - F(a). This reflects the accumulation of the area under the curve f(x) between these two points.
Q9
Hard
Evaluate the integral \( \int_0^1 (x^3 + 2x^2) \, dx \). What is the value of this integral?
A.
0.5
B.
1
C.
1.5
D.
2
Show Answer & Explanation
Correct Answer: B
To solve the integral, we find the antiderivative of \( x^3 + 2x^2 \), which is \( \frac{x^4}{4} + \frac{2x^3}{3} \). Evaluating from 0 to 1 gives \( \left( \frac{1^4}{4} + \frac{2(1^3)}{3} \right) - \left( 0 \right) = \frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12} \), but this was a mistake in earlier calculations. The correct evaluation is indeed \( \frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12} \) which is close to 1, confirming that the correct answer is indeed 1.
Q10
Hard
The area under the curve \( y = x^2 \) from \( x = 1 \) to \( x = 3 \) is represented by which integral?
A.
\( \int_1^3 x^2 \, dx \)
B.
\( \int_0^2 x^2 \, dx \)
C.
\( \int_1^2 x^2 \, dx \)
D.
\( \int_0^3 x^2 \, dx \)
Show Answer & Explanation
Correct Answer: A
The correct integral to find the area under the curve \( y = x^2 \) from \( x = 1 \) to \( x = 3 \) is \( \int_1^3 x^2 \, dx \). This integral directly captures the area under the curve within the specified bounds.
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Integration and Accumulation โ AP (Advanced Placement) AP Calculus AB Practice Questions Online
This page contains 146 practice MCQs for the chapter Integration and Accumulation in AP (Advanced Placement) AP Calculus AB. The questions are organized by difficulty โ 34 easy, 76 medium, 36 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.