Question 1

Which of the following integrals is considered 'impossible' to evaluate in terms of elementary functions?

Accepted Answer

$$\int e$$^{x^2}$$ dx$$

Question 2

Suppose you are given experimental data for a chemical reaction rate as a function of time, but no explicit formula. Which numerical integration method would be most appropriate to estimate the total amount of product formed?

Accepted Answer

Use the Trapezoidal Rule on the data points.

Question 3

Given $$f(x) = \sqrt{1 + x^3$} on$$ $$[0,2]$$, and using the Trapezoidal Rule with $$n=4$$, what is the value of $$\Delta x$$?

Accepted Answer

$$\Delta x = \frac{1}{2}$$

Question 4

Which formula correctly represents the Trapezoidal Rule for approximating $$\int_a^b f(x) dx$$ with $$n$$ subintervals?

Accepted Answer

$$T_n = \frac{\Delta x}{2} [f(x_0$) + 2f(x_1$) + 2f(x_2$) + \cdots + 2f(x_{n-1}) + f(x_n)]$$

Question 5

A pharmaceutical company wants to estimate the area under the curve $$y = e$$^{-x^2/2}$$ between x=a$ and x=b$ (as shown in the bell-shaped curve). Why can't they find an explicit formula for this area?

Accepted Answer

Because $$e^{-x^2/2}$$ does not have an elementary antiderivative.

Question 6

Which of the following statements about Simpson's Rule is TRUE?

Accepted Answer

Simpson's Rule requires an even number of subintervals.

Question 7

Given $$\$$int_0$$^\pi \sin(x) dx$$, which Simpson's Rule approximation with $$n=4$$ subintervals is correct?

Accepted Answer

$$S_4$$ = \frac{\pi}{12} [\sin(0) + 4\sin(\frac{\pi}{4}) + 2\sin(\frac{\pi}{2}) + 4\sin(\frac{3\pi}{4}) + \sin(\pi)]$$

Question 8

A medical researcher collects blood glucose levels at regular intervals over 8 hours. Which method would provide the most accurate estimate of the total glucose exposure if the data is smooth and $$n=8$$?

Accepted Answer

Simpson's Rule

Question 9

Which of the following is the correct error behavior as the number of subintervals $$n$$ increases for both the Trapezoidal and Simpson's Rule?

Accepted Answer

Both methods become more accurate as $$n$$ increases.

Question 10

Suppose you use the Trapezoidal Rule with $$n=4 to $$approximate $$\$$int_0$$^2 \sqrt{1 + x^3$} dx$$ and obtain $$T_4$$ \approx 3.283$$. If you double the number of subintervals to n=8$, what will happen to the approximation?

Accepted Answer

It will become closer to the exact value.

Question 11

Which of the following best describes the main difference between the Trapezoidal Rule and Simpson's Rule?

Accepted Answer

The Trapezoidal Rule uses straight lines, while Simpson's Rule uses parabolas to approximate the curve.

Question 12

If $$S_n is $$the Simpson's Rule approximation and $$T_n is $$the Trapezoidal Rule approximation for $$\int_a^b f(x) dx$$, which of the following is true?

Accepted Answer

$$S_n is $$generally more accurate than $$T_n$$ for the same $$n if$$ $$f(x) is $$smooth.

Question 13

A physicist wants to estimate the work done by a variable force $$F(x)$$ over an interval $$[a, b]$$ using only discrete measurements. Which method should they use?

Accepted Answer

Numerical integration, such as the Trapezoidal or Simpson's Rule.

Question 14

Which of the following is the correct formula for Simpson's Rule with $$n$$ even subintervals?

Accepted Answer

$$S_n = \frac{\Delta x}{3} [f(x_0$) + 4f(x_1$) + 2f(x_2$) + 4f(x_3$) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]$$

Question 15

If you use Simpson's Rule with $$n=4 to $$approximate $$\$$int_0$$^4 (\frac{1}{8}x^2$ + 1) dx$$, which weights are applied to the function values at the nodes?

Accepted Answer

1, 4, 2, 4, 1

Question 16

Which of the following is a real-world scenario where Simpson's Rule would be preferred over the Trapezoidal Rule?

Accepted Answer

Estimating the area under a smooth, curved temperature profile over a day using hourly measurements.

Question 17

Which of the following best explains why Simpson's Rule is more accurate for polynomials of degree three or less?

Accepted Answer

Because Simpson's Rule exactly integrates all polynomials up to degree three.

Techniques of Integration: Numerical Methods

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