Q: What is the formal definition of the definite integral $$\int_a^b f(x)\,dx$$?

$$\int_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^n f(x_i)\Delta x$$ where $$\Delta x = \frac{b-a}{n}$$ and $$x_i$$ are sample points.

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 amount of $$f(x)$$ over the interval $$[a, b]$$, found by summing infinitely many small products $$f(x_i)\Delta x.$$

Question 2

Suppose you want to estimate the area under $$y=f(x)$$ from $$x=a to$$ $$x=b. $$Which method is used to approximate this area before taking the limit?

Accepted Answer

Divide the interval into $$n$$ rectangles and sum their areas $$\sum_{i=1}^n f(x_i)\Delta x.$$

Question 3

What is the formal definition of the definite integral $$\int_a^b f(x)\,dx$$?

Accepted Answer

$$\int_a^b f(x)\,dx = \lim_{n	o\infty} \sum_{i=1}^n f(x_i)\Delta x$$ where $$\Delta x = \frac{b-a}{n}$$ and $$x_i$$ are sample points.

Question 4

Which statement about the Fundamental Theorem of Calculus (FTC) is correct?

Accepted Answer

It states that if $$F is an $$antiderivative of $$f$$, then $$\int_a^b f(x)\,dx = F(b) - F(a).$$

Question 5

Given $$f(x) is $$continuous on $$[a, b]$$, what is $$\int_a^b f(x)\,dx$$ according to the FTC?

Accepted Answer

$$F(b) - F(a)$$, where $$F is $$any antiderivative of $$f(x).$$

Question 6

If you want to find the area between two curves $$y=f(x)$$ and $$y=g(x) on$$ $$[a, b]$$, which formula should you use?

Accepted Answer

$$\int_a^b [f(x) - g(x)]\,dx$$ where $$f(x) is $$the upper function.

Question 7

Which of the following is NOT a typical application of definite integrals?

Accepted Answer

Determining the instantaneous rate of change at a point.

Question 8

To find the area between $$y=\sin x$$ and $$y=1-\sin x$$ for $$0 \leq x \leq \frac{\pi}{2}$$, which integral correctly represents the area?

Accepted Answer

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

Question 9

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

Accepted Answer

A negative value equal to the area below the $$x-$$axis between $$f(x)$$ and the $$x-$$axis.

Question 10

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

Accepted Answer

Differentiate both functions.

Question 11

Given $$y=x^2$ and y=x+2$, what is the area between these curves from x=0$ to x=2$?

Accepted Answer

$$\$$int_0$$^2 [(x+2) - x^2$] dx$$

Question 12

Which of the following is the correct Riemann sum for approximating $$\int_a^b f(x)\,dx$$?

Accepted Answer

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

Question 13

If $$\Delta x = \frac{b-a}{n}$$, what happens to the approximation of the area under $$f(x) as$$ $$n$$ increases?

Accepted Answer

The approximation becomes more accurate.

Question 14

Which integral represents the area between $$y=\frac{3}{x}$$ and $$x=7$$ for $$x=2 to$$ $$x=8$$?

Accepted Answer

$$\$$int_2$$^8 \frac{3}{x} dx$$

Question 15

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

Accepted Answer

It equals the net area between $$f(x)$$ and the $$x-$$axis on $$[a, b].$$

Question 16

Which of the following is a correct interpretation of $$\int_a^b f(x)\,dx in a $$real-world context?

Accepted Answer

The total distance traveled if $$f(x) is $$velocity over $$[a, b].$$

Question 17

If $$f(x) is $$the rate at which water flows into a tank (in liters per hour), what does $$\$$int_0$$^5 f(x)\,dx$$ represent?

Accepted Answer

The total amount of water added to the tank in 5 hours.

Definite Integrals and Area Between Curves

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