Lab 6: Student Assignment

Week 6, February 15

MAT 229, Spring 2021

Exercises to submit

Submit a Mathematica notebook with your results. For the hand calculations, report the results in the notebook as outlined in the questions.

Exercise 1

Consider the definite integral (∫_1^e)x (((ln(x)))^2)d x.

Compute this integral by hand. What technique(s) of integration did you use?

For any substitutions, say what your substitutions were.

For any integration by parts, say what your choices for u and d v are.

What is the result of your hand calculation?

Set up the summation to numerically integrate this using the midpoint rule with n=20, and have Mathematica evaluate the sum.

What is the maximum error in this estimation from the midpoint rule error estimate?

What is the actual error?

Exercise 2

Consider the definite integral (∫_0^2)(x^3)(Sqrt[4-(x^2)])d x.

Compute this integral by hand. What technique(s) of integration did you use?

For any substitutions, say what your substitutions were.

For any integration by parts, say what your choices for u and d v are.

What is the result of your hand calculation?

Estimate this integral using the left endpoint rule with n=50.

Estimate this integral using the right endpoint rule with n=50.

Estimate this integral using the trapezoid rule with n=50.

Exercise 3

Consider the definite integral (∫_0^10)(3sin((Sqrt[1+(x^2)]))+x)d x.

Estimate this integral using the trapezoid rule with n=100.

What is the maximum error in your estimate as given by the trapezoid rule error estimate?

Estimate this integral using the midpoint rule with n=100.

What is the maximum error in this estimate as given by the midpoint rule error estimate?

Estimate this integral using Simpson’s rule with n=100.

What is the maximum error in this estimate as given by the Simpson rule error estimate?