Computing the value of a definite integral using an antiderivative.
30 questions
AP Learning Target(s): LT 7: The Fundamental Theorem of Calculus and Definite Integrals
I.
Using the antiderivative of $$\cos(x)$$, we have $$\int_{0}^{\pi/2} \cos(x) dx = \sin(\pi/2) - \sin(0) = 1.$$
II.
If one mistakenly uses $$-\sin(x)$$ as the antiderivative, then $$\int_{0}^{\pi/2} \cos(x) dx$$ would be computed as $$-\sin(\pi/2) + \sin(0) = -1.$$
III.
A geometric interpretation of the area under $$\cos(x)$$ from 0 to $$\pi/2$$ also yields an area of 1.
Which of the statements correctly describe the evaluation of the integral $$\int_{0}^{\pi/2} \cos(x) dx$$?
I only
II only
I and III only
I, II, and III
Evaluate $$\int_{3}^{7} 5 dx$$.
$$30$$
$$15$$
$$25$$
$$20$$
Evaluate $$\int_{3}^{5} 4 dx$$.
$$4$$
$$8$$
$$2$$
$$12$$
Which property of definite integrals states that the integral of a constant function over an interval equals the product of the constant and the length of the interval?
Substitution property
Power rule
Mean Value Theorem
Constant property
Which of the following statements is always true regarding the definite integral of a non-negative function over a given interval?
The definite integral is always positive.
The definite integral is always zero.
The definite integral is non-zero.
The definite integral is always negative.
Which of the following is an example of correctly evaluating the definite integral $$\int_{0}^{2}4*x dx$$ using antiderivatives?
Evaluate by reversing the limits to get $$2*x^2$$ at 0 and 2, then subtract in reverse order to obtain -8.
Multiply the function value at the midpoint by the interval length: $$4*(1)*2=8$$.
Assume the antiderivative is $$F(x)=4*x^2$$ and compute $$4*2^2-4*0^2=16$$.
Correctly determine the antiderivative $$F(x)=2*x^2$$ and compute $$F(2)-F(0)=2*4-0=8$$.
The function $$f$$ is defined by $$f(x) = \begin{cases} \dfrac{x^2-4}{x-2} & \text{for } x \neq 2 \\ 5 & \text{for } x=2 \end{cases}$$. What is the value of $$\int_{1}^{3} f(x) dx$$?
14/3 + 5
11/3
The integral does not exist
8
For what values of $$x$$ is $$\displaystyle \int_{x}^{2} t^3 dt$$ equal to $$0$$?
$$2$$ only
$$-2$$ only
$$-2$$ and $$2$$ only
$$0$$ only
Evaluate $$\int_{1}^{9} \frac{1}{\sqrt{x}} dx$$.
$$\sqrt{9}-\sqrt{1}$$
$$4$$
$$2*(\sqrt{9}+\sqrt{1})$$
$$\frac{1}{2}(\sqrt{9}-\sqrt{1})$$
What is the value of $$\int_{1}^{4} (3x^2+5) dx$$?
84
80
75
78
Which of the following statements is true regarding the evaluation of the definite integral $$\int_{2}^{3} 2x dx$$?
The definite integral equals $$5$$.
The definite integral equals $$7$$ due to an incorrect subtraction.
The definite integral equals $$2$$ by mistakenly using only the coefficient.
The definite integral equals $$6$$ by miscomputing the antiderivative.
Which of the following statements correctly evaluates the definite integral $$\int_{1}^{4} 2x dx$$?
The definite integral equals $$7$$ by averaging the function values at the endpoints.
The definite integral equals $$30$$ because the integrand is mistakenly doubled.
The definite integral equals $$9$$ by evaluating $$2*x$$ at the midpoint of the interval.
The definite integral equals $$15$$ since the antiderivative is $$x^2$$ and $$4^2-1^2=15$$.
Evaluate $$\int_{0}^{3} (x+1)^2 dx$$.
$$21$$
$$20$$
$$18$$
$$14$$
Evaluate $$\int_{2}^{3} 2*x dx$$.
$$9$$
$$5$$
$$7$$
$$5 + C$$
What is the value of $$\int_{-1}^{1} x^2 dx$$?
$$\frac{2}{3}$$
$$\frac{1}{3}$$
$$\frac{2}{5}$$
$$\frac{4}{3}$$
Evaluate $$\int_{0}^{2} \left(\frac{1}{1+x^2}+\frac{x}{\sqrt{1+x}}\right) dx$$.
$$\tan^{-1}(2)-\frac{4}{3}$$
$$\tan^{-1}(2)+\frac{2}{3}$$
$$\tan^{-1}(2)+\frac{4}{3}$$
$$\tan^{-1}(x)\Big|_{0}^{2}+\frac{4}{3}$$
Evaluate $$\int_{1}^{4} \left(\frac{1}{2x}+\frac{1}{\sqrt{x}}\right)dx$$.
$$\frac{1}{2}\ln(2)+2$$
$$\ln(2)+2$$
$$2\ln(2)+2$$
$$\ln(2)+4$$
Evaluate the definite integral $$\int_{-1}^{2} (3*x+4) dx$$ using the Fundamental Theorem of Calculus.
Approximately $$16$$ due to rounding errors.
$$\frac{33}{2}$$
$$15$$
$$\frac{27}{2}$$
Evaluate $$\int_{2}^{3} 2x dx$$.
10
8
7
5
I.
$$\int_{0}^{2} 4*x^3 dx = 2^4 - 0^4 = 16$$
II.
$$\int_{0}^{2} 4*x^3 dx = \frac{2^4}{4} - \frac{0^4}{4} = 4$$
III.
$$\int_{0}^{2} 4*x^3 dx = 4*(2^3) - 4*(0^3) = 32$$
Which of the statements above is correct?
III only
II only
I only
I and II
Evaluate $$\int_{2}^{3} 2x dx$$.
$$4$$
$$5$$
$$7$$
$$6$$
Evaluate $$\int_{-1}^{2} (6x^2 + 4x) dx$$.
26
22
24
20
Compute the definite integral $$\int_{0}^{8} \frac{1}{2}*x^2 dx$$ using the Fundamental Theorem of Calculus.
$$\frac{512}{3}$$
$$\frac{256}{3}$$
$$\frac{128}{3}$$
Approximately $$\frac{256}{6}$$ due to a simplification error.
What is the value of the definite integral $$\displaystyle\int_{0}^{1} \left(e^x + x^2\right) dx$$?
$$e - 1$$
$$\displaystyle\frac{e}{3} + 1$$
$$e - \displaystyle\frac{2}{3}$$
$$e + \displaystyle\frac{1}{3}$$
What is the value of $$ \displaystyle\int_{0}^{4} \left(\sqrt{x} - 1\right) dx $$?
$$4$$
$$\displaystyle\frac{4}{3}$$
$$\displaystyle\frac{16}{3}$$
$$\displaystyle\frac{8}{3}$$
What is the value of $\displaystyle\int_{1}^{2} \left(\frac{1}{x} + x\right) dx$?
$$\displaystyle\frac{3}{2}$$
$$\ln(2) + \displaystyle\frac{3}{2}$$
$$\ln(2) + 1$$
$$2\ln(2) + \displaystyle\frac{1}{2}$$
What is the value of $$\int_{0}^{4} (3 + 4t) dt$$?
$42$
$39$
$44$
$46$
Which of the following correctly shows the evaluation of the definite integral that gives the area under the curve of $$f(x)=x^2$$ from $$x=0$$ to $$x=3$$?
$$x^2\Big|_{0}^{3}$$
$$\frac{x^3}{3} + C\Big|_{0}^{3}$$
$$\frac{x^3}{3}\Big|_{0}^{3} = \frac{3^3}{3} - \frac{0^3}{3}$$
$$\int_{0}^{3} x^2 dx + C$$
I.
$$\int 5 dx = 5x + C$$
II.
$$\int 5 dx = 5x$$
III.
$$\int 5 dx = x + 5 + C$$
Which of the statements above correctly represents the indefinite integral of the constant function 5?
III only
I and II
I only
II only
Given the function `F(x) = x⁴/4 - 3x² + 7x`, where `F'(x) = f(x)`, which of the following is `$$\int$$f(x)dx`?
x⁴/4 - 3x² + 7x
x⁴/4 - 3x² + 7x + C
x⁴/4 - 3x² + 7x - 5
x³ - 6x + 7