Question 1

Given the graph of $$y = x^3$ and the shaded region between x = 1$ and x = 5$, what is the area under the curve in this interval?

Accepted Answer

$$\displaystyle \$$int_1$$^5 x^3$ \, dx = \frac{1}{4}(5^4$ - 1^4$) = 156$$

Question 2

Given the graph of $$y = x^4$ and the shaded region between x = 1$ and x = 3$, what is the area under the curve in this interval?

Accepted Answer

$$\displaystyle \$$int_1$$^3 x^4$ \, dx = \frac{1}{5}(3^5$ - 1^5$) = 48.4$$

Question 3

Which of the following best describes the process for finding the area under a curve between two points $$a$$ and $$b$$?

Accepted Answer

Calculate the definite integral $$\displaystyle \int_a^b f(x) \, dx$$

Question 4

If $$f(x) = x^3$, what is the average value of f(x$$)$$ on the interval $$[1, 5]$$?

Accepted Answer

$$\displaystyle \frac{1}{5-1} \$$int_1$$^5 x^3$ \, dx = \frac{156}{4} = 39$$

Question 5

For the function $$f(x) = x^4$, what is the value of $$\displaystyle \$$int_1^3$$ $$x^4$$ \, dx$$?

Accepted Answer

$$48.4$$

Question 6

Which of the following statements is true about the area under $$y = x^3$ from x = 1$ to x = 5$?

Accepted Answer

The area is positive and equals $$156$$

Question 7

If $$f(x) = x^4$, what is the antiderivative F(x$$)$$?

Accepted Answer

$$F(x) = \frac{1}{5}x^5$ + C$$

Question 8

Which of the following best represents the shaded region in the first image?

Accepted Answer

The area under $$y = x^3$ from x = 1$ to x = 5$

Question 9

Which of the following best represents the shaded region in the second image?

Accepted Answer

The area under $$y = x^4$ from x = 1$ to x = 3$

Question 10

If $$f(x) = x^3$, what is the definite integral from x = 1$ to x = 5$?

Accepted Answer

$$156$$

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