Q: If $$f(x) is $$continuous on $$[a, b]$$, which of the following statements is true about $$\int_a^b f(x)\,dx$$?

It can be computed as the limit of Riemann sums as $$n \to \infty.$$

Q: Which of the following is the correct limit definition of the definite integral $$\int_a^b f(x)\,dx$$?

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

Question 1

Which of the following best describes the meaning of a definite integral $$\int_a^b f(x)\,dx$$?

Accepted Answer

It represents the total accumulation of $$f(x)$$ over $$[a, b]$$, often interpreted as the area under the curve $$y=f(x)$$ from $$x=a to$$ $$x=b.$$

Question 2

Suppose you want to estimate the area under $$y=f(x)$$ from $$x=a to$$ $$x=b$$ using $$n$$ rectangles of equal width. What is the width $$\Delta x of $$each rectangle?

Accepted Answer

$$\Delta x = \frac{b-a}{n}$$

Question 3

Which of the following expressions represents a right-endpoint Riemann sum for $$f(x) on$$ $$[a, b]$$ with $$n$$ subintervals?

Accepted Answer

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

Question 4

The Fundamental Theorem of Calculus states that if $$F is an $$antiderivative of $$f on$$ $$[a, b]$$, then:

Accepted Answer

$$\int_a^b f(x)\,dx = F(b) - F(a)$$

Question 5

Which of the following is the correct setup for the area between the curves $$y=f(x)$$ and $$y=g(x) on$$ $$[a, b]$$, where $$f(x) \geq g(x)$$?

Accepted Answer

$$\int_a^b [f(x) - g(x)]\,dx$$

Question 6

A biologist measures the rate at which a chemical is produced in a reaction as $$r(t) mg/$$min over $$0 \leq t \leq 10$$ minutes. What does $$\$$int_0$$^{10} r(t)\,dt$$ represent?

Accepted Answer

The total amount of chemical produced in 10 minutes.

Question 7

Given $$\$$int_1$$^4 (2x+1)\,dx$$, what is the value of the definite integral?

Accepted Answer

15

Question 8

If $$f(x) is $$continuous on $$[a, b]$$, which of the following statements is true about $$\int_a^b f(x)\,dx$$?

Accepted Answer

It can be computed as the limit of Riemann sums as $$n 	o \infty.$$

Question 9

A physics student wants to find the work done by a variable force $$F(x)$$ moving an object from $$x=a to$$ $$x=b. $$Which integral should they use?

Accepted Answer

$$\int_a^b F(x)\,dx$$

Question 10

Which of the following is a correct interpretation of $$\$$int_0$$^2 x^2$\,dx$$?

Accepted Answer

The area under $$y=x^2$ from x=0$ to x=2$.

Question 11

If $$\$$int_1$$^3 f(x)\,dx = 5$$ and $$\$$int_3$$^5 f(x)\,dx = 7$$, what is $$\$$int_1$$^5 f(x)\,dx$$?

Accepted Answer

12

Question 12

Which of the following is the correct setup for the area between $$y=\sin x$$ and $$y=1-\sin x on$$ $$[0, \frac{\pi}{2}]$$?

Accepted Answer

$$\$$int_0$$^{\frac{\pi}{2}} [\sin x - (1-\sin x)]\,dx$$

Question 13

A region is bounded by $$y=x^2$, y=0$, and x=2$. What is the area of this region?

Accepted Answer

$$\$$int_0$$^2 x^2$\,dx$$

Question 14

Which of the following is NOT a possible application of definite integrals as described in the study materials?

Accepted Answer

Determining the slope of a tangent line.

Question 15

If $$f(x) is $$negative on $$[a, b]$$, what does $$\int_a^b f(x)\,dx$$ represent?

Accepted Answer

A negative value, representing the net area below the $$x-$$axis.

Question 16

Which of the following is the correct limit definition of the definite integral $$\int_a^b f(x)\,dx$$?

Accepted Answer

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

Question 17

If $$\$$int_0$$^3 f(x)\,dx = 8$$ and $$\$$int_3$$^5 f(x)\,dx = -2$$, what is $$\$$int_0$$^5 f(x)\,dx$$?

Accepted Answer

6

Question 18

Which of the following steps is NOT part of the process for finding the area between two curves?

Accepted Answer

Differentiate both functions.

Applications of Integration: Area and the Definite Integral

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