Q: Which of the following best describes the process of finding the exact area under a curve using Riemann sums?

Take the limit as the number of rectangles $$n \to \infty.$$

Question 1

Which of the following statements best describes the family of curves shown in the first image, where $$y = x^3$ + C$$ and $$C$$ varies?

Accepted Answer

Each curve represents a different solution to the differential equation $$\frac{dy}{dx} = 3x^2$.

Question 2

Given the point $$(1, -1) on $$the curve $$y = x^3$ + C$$, what is the value of $$C$$?

Accepted Answer

$$C = -2$$

Question 3

In the second image, rectangles are used to estimate the area under $$y = 1 - x^2$ from x = 0$ to x = 1$. Which method is being used?

Accepted Answer

Left endpoint Riemann sum

Question 4

For the function $$y = 1 - x^2$ on $$[0,1]$$, which of the following statements is true about the left endpoint Riemann sum estimate?

Accepted Answer

It is an underestimate because the function is decreasing.

Question 5

In the third image, rectangles are drawn under the curve $$y = x^2$ from x = 0$ to x = 3$. What is the width $$\Delta x$$ of each rectangle if there are 6 rectangles?

Accepted Answer

$$\Delta x = 0.5$$

Question 6

Which Riemann sum method is illustrated in the third image, where the heights of the rectangles are determined by the function value at the left endpoint of each interval?

Accepted Answer

Left endpoint Riemann sum

Question 7

As the number of rectangles increases in the Riemann sum approximation for the area under a curve, what happens to the accuracy of the estimate?

Accepted Answer

The estimate becomes more accurate and approaches the exact area.

Question 8

In the fourth image, rectangles are drawn under the curve $$y = 1 - x^2$ from x = 0$ to x = 1$. If the number of rectangles is increased from 4 to 12 to 24, what is the effect on the approximation?

Accepted Answer

The approximation becomes closer to the true area under the curve.

Question 9

Which of the following best describes the process of finding the exact area under a curve using Riemann sums?

Accepted Answer

Take the limit as the number of rectangles $$n 	o \infty.$$

Question 10

In the fifth image, the area between $$y = 9x^2$ - 16x + 4$$ and the x-axis is shaded. Which integral represents the total (unsigned) area between the curve and the x-axis from $$x = c_1$ to x$$ = $$c_2$$?

Accepted Answer

$$\int_{c_1$}^{c_2$} |9x^2$ - 16x + 4| dx$$

Question 11

Which property of definite integrals allows you to split the integral $$\int_{a}^{c} f(x) dx$$ into $$\int_{a}^{b} f(x) dx + \int_{b}^{c} f(x) dx$$?

Accepted Answer

$$\int_{a}^{c} f(x) dx = \int_{a}^{b} f(x) dx + \int_{b}^{c} f(x) dx$$

Question 12

If $$F(x) is an $$antiderivative of $$f(x)$$, what is the most general antiderivative of $$f(x)$$?

Accepted Answer

$$F(x) + C$$

Question 13

Which of the following is the correct formula for the definite integral of $$x^n$$ from $$a to$$ $$b ($$where $$n 
eq -1$$)?

Accepted Answer

$$\int_{a}^{b} x^n dx = \frac{b$$^{n+1}$$}{n+1} - \frac{a$$^{n+1}$$}{n+1}$$

Question 14

Which of the following is the correct application of the Fundamental Theorem of Calculus, Part II?

Accepted Answer

$$\int_{a}^{b} f(x) dx = F(b) - F(a)$$, where $$F(x) is $$any antiderivative of $$f(x).$$

Question 15

If $$s(t) is $$the position function and $$v(t) is $$the velocity function, which integral gives the displacement from $$t_0 to$$ $$t_1$$?

Accepted Answer

$$\int_{t_0$}^{t_1$} v(t) dt = s(t_1$) - s(t_0$)$$

Question 16

Which substitution would be most appropriate for evaluating $$\int x^2$ e$$^{x^3}$$ dx$$?

Accepted Answer

Let $$u = x^3$, so du$$ = $$3x^2 dx$

Question 17

If $$y = f(x)$$ and $$y = g(x)$$ are continuous and $$f(x) \geq g(x) on$$ $$[a, b]$$, what is the area between the curves?

Accepted Answer

$$\int_{a}^{b} [f(x) - g(x)] dx$$

Calculus I: Applications of Derivatives and Introduction to Integration

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