Fundamental Theorem of Calculus: Definite Intervals - AP Calculus AB
Card 1 of 30
What is an antiderivative of $f(x) = 3x^2$?
What is an antiderivative of $f(x) = 3x^2$?
Tap to reveal answer
$F(x) = x^3 + C$. Power rule: increase exponent by 1, divide by new exponent.
$F(x) = x^3 + C$. Power rule: increase exponent by 1, divide by new exponent.
← Didn't Know|Knew It →
Calculate $\int_1^3 (2x^2 - x),dx$ using the Fundamental Theorem.
Calculate $\int_1^3 (2x^2 - x),dx$ using the Fundamental Theorem.
Tap to reveal answer
$\frac{26}{3}$. Use antiderivative $\frac{2x^3}{3} - \frac{x^2}{2}$, evaluate at bounds.
$\frac{26}{3}$. Use antiderivative $\frac{2x^3}{3} - \frac{x^2}{2}$, evaluate at bounds.
← Didn't Know|Knew It →
Evaluate $\int_{-2}^2 (x^3 + 2x),dx$ using symmetry properties.
Evaluate $\int_{-2}^2 (x^3 + 2x),dx$ using symmetry properties.
Tap to reveal answer
$0$. Both functions are odd, so integral over symmetric interval is zero.
$0$. Both functions are odd, so integral over symmetric interval is zero.
← Didn't Know|Knew It →
Evaluate $\int_0^4 (3x^2 + 2),dx$ using the Fundamental Theorem.
Evaluate $\int_0^4 (3x^2 + 2),dx$ using the Fundamental Theorem.
Tap to reveal answer
- Use antiderivative $x^3 + 2x$, evaluate at bounds $4$ and $0$.
- Use antiderivative $x^3 + 2x$, evaluate at bounds $4$ and $0$.
← Didn't Know|Knew It →
State the Fundamental Theorem of Calculus, Part 1.
State the Fundamental Theorem of Calculus, Part 1.
Tap to reveal answer
If $F$ is an antiderivative of $f$ on $[a,b]$, then $\int_a^b f(x),dx = F(b) - F(a)$. This allows calculation of definite integrals using antiderivatives.
If $F$ is an antiderivative of $f$ on $[a,b]$, then $\int_a^b f(x),dx = F(b) - F(a)$. This allows calculation of definite integrals using antiderivatives.
← Didn't Know|Knew It →
Determine $\int_1^4 \frac{1}{x},dx$.
Determine $\int_1^4 \frac{1}{x},dx$.
Tap to reveal answer
$\ln(4)$. Antiderivative of $\frac{1}{x}$ is $\ln(x)$.
$\ln(4)$. Antiderivative of $\frac{1}{x}$ is $\ln(x)$.
← Didn't Know|Knew It →
For $F(x) = \int_0^x e^t,dt$, find $F'(x)$.
For $F(x) = \int_0^x e^t,dt$, find $F'(x)$.
Tap to reveal answer
$e^x$. By Part 2 of FTC, the derivative equals the integrand.
$e^x$. By Part 2 of FTC, the derivative equals the integrand.
← Didn't Know|Knew It →
For $F(x) = \int_0^x ,\sin(t),dt$, find $F'(x)$.
For $F(x) = \int_0^x ,\sin(t),dt$, find $F'(x)$.
Tap to reveal answer
$\sin(x)$. By Part 2 of FTC, derivative equals the integrand.
$\sin(x)$. By Part 2 of FTC, derivative equals the integrand.
← Didn't Know|Knew It →
What does the second part of the Fundamental Theorem allow us to compute?
What does the second part of the Fundamental Theorem allow us to compute?
Tap to reveal answer
The derivative of an integral function. It gives the rate of change of the area function.
The derivative of an integral function. It gives the rate of change of the area function.
← Didn't Know|Knew It →
Determine $\int_0^2 x^2,dx$ using an antiderivative.
Determine $\int_0^2 x^2,dx$ using an antiderivative.
Tap to reveal answer
$\frac{8}{3}$. Use antiderivative $\frac{x^3}{3}$, evaluate at bounds.
$\frac{8}{3}$. Use antiderivative $\frac{x^3}{3}$, evaluate at bounds.
← Didn't Know|Knew It →
Calculate $\int_1^3 (2x^2 - x),dx$ using the Fundamental Theorem.
Calculate $\int_1^3 (2x^2 - x),dx$ using the Fundamental Theorem.
Tap to reveal answer
$\frac{26}{3}$. Use antiderivative $\frac{2x^3}{3} - \frac{x^2}{2}$, evaluate at bounds.
$\frac{26}{3}$. Use antiderivative $\frac{2x^3}{3} - \frac{x^2}{2}$, evaluate at bounds.
← Didn't Know|Knew It →
What is the integral of $f(x) = e^x$ over $[0, 1]$?
What is the integral of $f(x) = e^x$ over $[0, 1]$?
Tap to reveal answer
$e - 1$. Antiderivative of $e^x$ is $e^x$; evaluate at bounds.
$e - 1$. Antiderivative of $e^x$ is $e^x$; evaluate at bounds.
← Didn't Know|Knew It →
What is a definite integral's geometric interpretation?
What is a definite integral's geometric interpretation?
Tap to reveal answer
Net area between the curve and the x-axis over $[a, b]$. Represents the signed area between curve and x-axis.
Net area between the curve and the x-axis over $[a, b]$. Represents the signed area between curve and x-axis.
← Didn't Know|Knew It →
Find the derivative of $F(x) = \int_2^x \ln(t),dt$.
Find the derivative of $F(x) = \int_2^x \ln(t),dt$.
Tap to reveal answer
$\ln(x)$. By Part 2 of FTC, the derivative equals the integrand.
$\ln(x)$. By Part 2 of FTC, the derivative equals the integrand.
← Didn't Know|Knew It →
Find $\int_0^2 (4x^3 + x),dx$ using an antiderivative.
Find $\int_0^2 (4x^3 + x),dx$ using an antiderivative.
Tap to reveal answer
- Use antiderivative $x^4 + \frac{x^2}{2}$, evaluate at bounds.
- Use antiderivative $x^4 + \frac{x^2}{2}$, evaluate at bounds.
← Didn't Know|Knew It →
State the Fundamental Theorem of Calculus, Part 2.
State the Fundamental Theorem of Calculus, Part 2.
Tap to reveal answer
If $f$ is continuous on $[a,b]$, then $F(x) = \int_a^x f(t),dt$ is differentiable and $F'(x) = f(x)$. Shows that differentiation and integration are inverse operations.
If $f$ is continuous on $[a,b]$, then $F(x) = \int_a^x f(t),dt$ is differentiable and $F'(x) = f(x)$. Shows that differentiation and integration are inverse operations.
← Didn't Know|Knew It →
What is the definite integral of a constant $c$ over an interval $[a, b]$?
What is the definite integral of a constant $c$ over an interval $[a, b]$?
Tap to reveal answer
$c(b-a)$. A constant function integrates to constant times interval length.
$c(b-a)$. A constant function integrates to constant times interval length.
← Didn't Know|Knew It →
Compute $\int_{-1}^1 x^3,dx$ using symmetry properties.
Compute $\int_{-1}^1 x^3,dx$ using symmetry properties.
Tap to reveal answer
- Odd function over symmetric interval gives zero area.
- Odd function over symmetric interval gives zero area.
← Didn't Know|Knew It →
Evaluate $\int_0^1 2x,dx$ using the Fundamental Theorem of Calculus.
Evaluate $\int_0^1 2x,dx$ using the Fundamental Theorem of Calculus.
Tap to reveal answer
- Find antiderivative $x^2$, then evaluate $1^2 - 0^2 = 1$.
- Find antiderivative $x^2$, then evaluate $1^2 - 0^2 = 1$.
← Didn't Know|Knew It →
Determine $\int_1^3 x^2,dx$ using an antiderivative.
Determine $\int_1^3 x^2,dx$ using an antiderivative.
Tap to reveal answer
$26/3$. Use antiderivative $\frac{x^3}{3}$, then evaluate at bounds.
$26/3$. Use antiderivative $\frac{x^3}{3}$, then evaluate at bounds.
← Didn't Know|Knew It →
What does the Fundamental Theorem of Calculus connect?
What does the Fundamental Theorem of Calculus connect?
Tap to reveal answer
It connects differentiation and integration. They are inverse operations of each other.
It connects differentiation and integration. They are inverse operations of each other.
← Didn't Know|Knew It →
Evaluate $\int_0^3 (x^2 + x + 1),dx$ using the Fundamental Theorem.
Evaluate $\int_0^3 (x^2 + x + 1),dx$ using the Fundamental Theorem.
Tap to reveal answer
- Use antiderivative $\frac{x^3}{3} + \frac{x^2}{2} + x$, evaluate at bounds.
- Use antiderivative $\frac{x^3}{3} + \frac{x^2}{2} + x$, evaluate at bounds.
← Didn't Know|Knew It →
Find $\int_0^4 (x^2 - 2),dx$ using the Fundamental Theorem.
Find $\int_0^4 (x^2 - 2),dx$ using the Fundamental Theorem.
Tap to reveal answer
$\frac{32}{3}$. Use antiderivative $\frac{x^3}{3} - 2x$, evaluate at bounds.
$\frac{32}{3}$. Use antiderivative $\frac{x^3}{3} - 2x$, evaluate at bounds.
← Didn't Know|Knew It →
Determine $\int_0^1 (5x^2 - 3x),dx$ using an antiderivative.
Determine $\int_0^1 (5x^2 - 3x),dx$ using an antiderivative.
Tap to reveal answer
$\frac{1}{3}$. Use antiderivative $\frac{5x^3}{3} - \frac{3x^2}{2}$, evaluate at bounds.
$\frac{1}{3}$. Use antiderivative $\frac{5x^3}{3} - \frac{3x^2}{2}$, evaluate at bounds.
← Didn't Know|Knew It →
Evaluate $\int_0^1 (4x^3 - x),dx$ using the Fundamental Theorem.
Evaluate $\int_0^1 (4x^3 - x),dx$ using the Fundamental Theorem.
Tap to reveal answer
$\frac{3}{4}$. Use antiderivative $x^4 - \frac{x^2}{2}$, evaluate at bounds.
$\frac{3}{4}$. Use antiderivative $x^4 - \frac{x^2}{2}$, evaluate at bounds.
← Didn't Know|Knew It →
What is the result of $\int_0^2 1,dx$?
What is the result of $\int_0^2 1,dx$?
Tap to reveal answer
- Integral of constant $1$ over interval length $2$.
- Integral of constant $1$ over interval length $2$.
← Didn't Know|Knew It →
Find $\int_0^1 (x^2 + 2x),dx$ using an antiderivative.
Find $\int_0^1 (x^2 + 2x),dx$ using an antiderivative.
Tap to reveal answer
$\frac{7}{3}$. Use antiderivative $\frac{x^3}{3} + x^2$, evaluate at bounds.
$\frac{7}{3}$. Use antiderivative $\frac{x^3}{3} + x^2$, evaluate at bounds.
← Didn't Know|Knew It →
What is the integral of $f(x) = \cos(x)$ over $[0, \pi]$?
What is the integral of $f(x) = \cos(x)$ over $[0, \pi]$?
Tap to reveal answer
- Antiderivative is $\sin(x)$; $\sin(\pi) - \sin(0) = 0$.
- Antiderivative is $\sin(x)$; $\sin(\pi) - \sin(0) = 0$.
← Didn't Know|Knew It →
What is the integral of $f(x) = 1/x$ over $[1, e]$?
What is the integral of $f(x) = 1/x$ over $[1, e]$?
Tap to reveal answer
- Antiderivative of $\frac{1}{x}$ is $\ln(x)$; $\ln(e) - \ln(1) = 1$.
- Antiderivative of $\frac{1}{x}$ is $\ln(x)$; $\ln(e) - \ln(1) = 1$.
← Didn't Know|Knew It →
Calculate $\int_0^1 (x^3 + 1),dx$ using the Fundamental Theorem.
Calculate $\int_0^1 (x^3 + 1),dx$ using the Fundamental Theorem.
Tap to reveal answer
$\frac{5}{4}$. Use antiderivative $\frac{x^4}{4} + x$, evaluate at bounds.
$\frac{5}{4}$. Use antiderivative $\frac{x^4}{4} + x$, evaluate at bounds.
← Didn't Know|Knew It →