AP (Advanced Placement) · Calculus Ab

Integration

68 practice questions, each with the answer and an explanation.

Practise Integration now

Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 68 questions

Evaluate $\int_{0}^{1} 6 x^{2} d x$ .
Evaluate $\int_{0}^{1} 4 x^{1} d x$ .
Evaluate $\int_{0}^{2} 5 x^{3} d x$ .
Evaluate $\int_{0}^{3} 4 x^{1} d x$ .
Evaluate $\int_{0}^{4} 5 x^{1} d x$ .
Evaluate $\int_{0}^{1} 5 x^{2} d x$ .
Evaluate $\int_{0}^{2} 4 x^{3} d x$ .
Evaluate $\int_{0}^{4} 6 x^{1} d x$ .
Evaluate $\int_{0}^{2} 3 x^{3} d x$ .
Evaluate $\int_{0}^{3} 4 x^{3} d x$ .
Evaluate $\int_{0}^{1} 5 x^{3} d x$ .
Evaluate $\int_{0}^{4} 4 x^{3} d x$ .
Evaluate $\int_{0}^{1} 3 x^{2} d x$ .
Evaluate $\int_{0}^{2} 3 x^{2} d x$ .
Evaluate $\int_{0}^{3} 6 x^{1} d x$ .
Evaluate $\int_{0}^{2} 2 x^{3} d x$ .
Evaluate $\int_{0}^{3} 3 x^{1} d x$ .
Evaluate $\int_{0}^{3} 5 x^{2} d x$ .
Evaluate $\int_{0}^{2} 5 x^{1} d x$ .
Evaluate $\int_{0}^{3} 6 x^{2} d x$ .
Evaluate $\int_{0}^{2} 2 x^{1} d x$ .
Evaluate $\int_{0}^{1} 4 x^{3} d x$ .
What is $\int_{0}^{1} x^{3} 1 + x^{2} d x$ ?
What is $\int_{0}^{π /4} tan^{3} x d x$ ?
What is the area of the region enclosed by the graphs of $y = x^{2}$ and $y = 2 x - x^{2}$ ?
The region in the first quadrant bounded by $y = x$ and $y = x^{2}$ is revolved about the $x$ -axis. What is the volume of the solid generated?
What is the average value of $f (x) = x x^{2} + 9$ on the interval $[0, 4]$ ?
A particle moves along a line with velocity $v (t) = t^{2} - 4 t + 3$ for $0 \leq t \leq 4$ . What is the total distance traveled by the particle?
What is $\int_{0}^{π /2} \frac{sin x cos x}{1 + sin ^{2} x} d x$ ?
The function $y (x)$ satisfies $\frac{d y}{d x} = x y^{2}$ with $y (0) = 1$ . What is $y (1)$ ?
Let $g (x) = \int_{0}^{x} (t^{2} - 4) d t$ . What is the maximum value of $g$ on the interval $[0, 5]$ ?
A function $f$ has values $f (0) = 3$ , $f (2) = 5$ , $f (5) = 4$ and $f (9) = 8$ . Using the trapezoidal rule with the three subintervals $[0, 2]$ , $[2, 5]$ and $[5, 9]$ , what is the approximation to…
Evaluate the definite integral: $\int_{0}^{3} (3 x^{2} - 2 x + 1) d x$ .
Evaluate the indefinite integral: $\int \frac{2 x}{x ^{2} + 1} d x$ .
Evaluate the definite integral: $\int_{0}^{π /2} cos (x) e^{s i n (x)} d x$ .
What is the average value of the function $f (x) = 4 - x^{2}$ on the interval $[0, 3]$ ?
Find the area of the region bounded by $y = x^{2}$ and $y = 2 x$ .
Find the volume of the solid generated by revolving the region bounded by $y = x$ , $y = 0$ , and $x = 4$ about the $x$ -axis.
Evaluate the definite integral: $\int_{1}^{e} \frac{( l n x ) ^{2}}{x} d x$ .
Evaluate the integral: $\int \frac{1}{1 - 9 x ^{2}} d x$ .
If $\int_{1}^{5} f (x) d x = 12$ and $\int_{3}^{5} f (x) d x = 7$ , what is $\int_{1}^{3} [2 f (x) + 1] d x$ ?
A left Riemann sum with $3$ subintervals of equal length is used to approximate $\int_{1}^{7} x^{2} d x$ . What is the value of this approximation?
Evaluate the definite integral: $\int_{0}^{1} \frac{1}{1 + x ^{2}} d x$ .
Solve the differential equation $\frac{d y}{d x} = 3 x^{2} y$ with the initial condition $y (0) = 4$ .
Evaluate the integral: $\int x x - 1 d x$ .
Evaluate $\int_{0}^{2} ∣2 x - 2∣ d x$ .
If $F (x) = \int_{2}^{x} \frac{1}{l n ( t )} d t$ for $x > 1$ , what is $F^{'} (e)$ ?
Find the total area enclosed between the curve $y = 4 - x^{2}$ and the $x$ -axis on $[- 2, 2]$ .
Evaluate $\int_{0}^{1} x^{2} (1 - x^{3})^{4} d x$ .
The base of a solid is the region in the first quadrant bounded by $y = 1 - x^{2}$ , the $x$ -axis, and the $y$ -axis. Cross sections perpendicular to the $x$ -axis are squares. Find the volume of the solid…
Evaluate the definite integral: $\int_{1}^{4} \frac{1}{2 x} d x$ .
If $\int_{0}^{k} 2 x d x = 16$ and $k > 0$ , what is the value of $k$ ?
Evaluate $\int_{0}^{3} 4 x^{2} d x$ .
Evaluate $\int_{0}^{4} 1 x^{1} d x$ .
What is the value of $\int_{0}^{π} sin^{2} (x) d x$ ?
Let $F (x) = \int_{1}^{x^{3}} t^{2} + 7 d t$ . What is $F^{'} (2)$ ?
What is the area of the region in the first quadrant enclosed by $y = x$ , the line $x = 4$ , and the $x$ -axis?
Evaluate $\int_{0}^{1} \frac{2 x}{x ^{2} + 1} d x$ .
The region bounded by $y = x^{2}$ and $y = 2 x$ is revolved around the $x$ -axis. What is the volume of the resulting solid?
What is the average value of $f (x) = 3 x^{2} - 2 x + 1$ on the interval $[1, 4]$ ?
Evaluate $\int_{0}^{3} \frac{1}{x + 1} d x$ .
What is the total area between the curve $y = cos (x)$ and the $x$ -axis on the interval $[0, π]$ ?
Evaluate $\int_{0}^{π /4} sec^{2} (x) e^{t a n (x)} d x$ .
A function $y (t)$ satisfies the differential equation $\frac{d y}{d t} = 2 y$ with initial condition $y (0) = 5$ . What is $y (3)$ ?
The base of a solid is the region bounded by $y = 1 - x^{2}$ and the $x$ -axis. Cross-sections perpendicular to the $x$ -axis are squares. What is the volume of the solid?
Evaluate $\int_{1}^{e} \frac{( ln x ) ^{2}}{x} d x$ .
Using a trapezoidal sum with three equal subintervals, what is the approximation of $\int_{0}^{6} f (x) d x$ for the data points $(0, 2), (2, 5), (4, 7), (6, 10)$ ?
A particle moves with acceleration $a (t) = 6 t - 4$ . If $v (0) = 3$ and $s (0) = 2$ , what is the position function $s (t)$ ?

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