Q: Which of the following is the correct limit definition of the definite integral of $$f(x)$$ from $$a to$$ $$b$$?

$$ \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x $$

Question 1

Given the velocity function $$v(t) of an $$object moving along a line, as shown in the graph, which of the following best describes the total displacement of the object from $$t = 0 to$$ $$t = 8$$ seconds?

Accepted Answer

The total displacement is the area under the curve above the $$t-$$axis minus the area below the $$t-$$axis.

Question 2

Which of the following is the correct limit definition of the definite integral of $$f(x)$$ from $$a to$$ $$b$$?

Accepted Answer

$$ \lim_{n 	o \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x $$

Question 3

Using the Fundamental Theorem of Calculus, evaluate $$ \int_{0}^{1} x^2$ dx .$$

Accepted Answer

$$ \frac{1}{3} $$

Question 4

Which of the following expressions represents the derivative with respect to $$x of$$ $$ \int_{0}^{x} \sqrt{1 + t^2$} dt $$?

Accepted Answer

$$ \sqrt{1 + x^2$} $$

Question 5

Evaluate $$ \int_{0}^{1} x^3$ - 7x^2$ + x dx .$$

Accepted Answer

$$ -\frac{19}{12} $$

Question 6

Which of the following is the correct antiderivative of $$ e$$^{x^2}$$ $$ with respect to $$x$$?

Accepted Answer

There is no elementary antiderivative for $$ e$$^{x^2}$$ .$$

Question 7

Evaluate $$ \int \frac{\cos(2x)}{1 + \sin^2$(x)} dx $$ given $$ \sin(2x) = 2 \sin(x) \cos(x) .$$

Accepted Answer

$$ \arctan(\sin(x)) + C $$

Question 8

A car's velocity is given by $$ v(t) = 3t^2$ - 2t + 1 . $$What is the total distance traveled from $$ t = 0  to$$ $$ t = 2 $$ seconds?

Accepted Answer

$$ \int_{0}^{2} |3t^2$ - 2t + 1| dt $$

Question 9

Which of the following statements about the definite integral $$ \int_{a}^{b} f(x) dx  is $$true?

Accepted Answer

It represents the net area between the graph of $$ f(x) $$ and the $$x-$$axis from $$ x = a  to$$ $$ x = b .$$

Question 10

If $$ F(x)  is an $$antiderivative of $$ f(x) $$, which of the following is true according to the Fundamental Theorem of Calculus?

Accepted Answer

$$ \int_{a}^{b} f(x) dx = F(b) - F(a) $$

Question 11

Which of the following is a correct application of the substitution method for $$ \int 2x \cos(x^2$) dx $$?

Accepted Answer

Let $$ u = x^2$ $$, then $$ du = 2x dx $$, so $$ \int 2x \cos(x^2$) dx = \int \cos(u) du $$

Question 12

Which of the following is the correct interpretation of $$ \int_{0}^{T} v(t) dt  in $$the context of physics?

Accepted Answer

It represents the displacement of an object over the time interval $$ [0, T] .$$

Question 13

Suppose $$ f(x)  is $$continuous on $$ [a, b] . $$Which of the following is always true about $$ \int_{a}^{b} f(x) dx $$?

Accepted Answer

It exists and is finite.

Question 14

Which of the following is the correct derivative of $$ \int_{0}^{x} e$$^{t^2}$$ dt $$ with respect to $$x$$?

Accepted Answer

$$ e$$^{x^2}$$ $$

Question 15

A medication is administered at a rate $$ r(t) = 5e$$^{-0.2t}$$  mg/$$hour. What is the total amount administered over 10 hours?

Accepted Answer

$$ \int_{0}^{10} 5e$$^{-0.2t}$$ dt $$

Question 16

Which of the following is the correct evaluation of $$ \int_{0}^{\pi} \sin(x) dx $$?

Accepted Answer

$$ 0 $$

Question 17

If $$ f(x)  is an $$odd function, which of the following is true about $$ \int_{-a}^{a} f(x) dx $$?

Accepted Answer

$$ 0 $$

Question 18

Which of the following is the correct evaluation of $$ \int_{0}^{1} \frac{1}{1 + x^2$} dx $$?

Accepted Answer

$$ \arctan(1) - \arctan(0) $$

Calculus II: Integration and Applications – Study Notes

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