Question

True or False Directions: Determine whether each statement below is correct or incorrect. Justify your response with a calculation, a description using complete English sentences, or a combination of both. If a statement is incorrect, make sure to include what the error is and state what the student should have done to evaluate the integral correctly in your explanation. All calculations must be done by hand and proper notation must be used. Correct responses with insufficient justification or that rely on technology will earn no credit.

A. [8 pts) Morgan claims that the improper integral * 4.re -2- dr converges to 1. esca 8
B. (8 pts] Avery is asked to compute dr and provides the following argument. (3 - 4.) 8 (3 - 4.0) (3 - 4.r) (-1) (3) lo dr =

Answer 1

(A) true.

(B). False.

Explanation:

(A). F(x) = 4xe^-2x.

Let's make the assumption that 2x = b -----------------------(1).

Therefore, taking the differentiation with respect to x, we have;

2x dx = db.

The next thing thing to do is to integrate, taking the upper limit to be infinity and the lower limit to be zero:

∫( b e^-b db).

Changing the Lim. infinity = 2.

= ∫( b e^-b db). ----------------------------(2).

The step (2) above can be solve with Integrations by part;

Lim. 2 ---> infinity [ b e^-b + ∫( 1 e^-b db|

( | = Upper boundary = 2 and the lower boundary = 0).

Lim. 2 ----> infinity [ 2 e^-2 - 0] - lim. 2 --> infinity [ 0 - 1].

Lim 2 ---> infinity [ 2/e^2 + 1].

Let 2/e^2 = j.

The, lim 2 ---> infinity j + 1.

To solve this, there is need to make use of L'hospital rule,

j = Lim 2 ---> infinity [ 1 /e^2] = 0.

Thus, j = 0. And 0 + 1 = 1.

(PROVED TO BE TRUE).

(B). Taking limit from 1 (upper) to 0( lower). Assuming that b = 3 - 4x.

Therefore, -4dx = db.

∫( 8 /(3 - 4x)^3 dx.

Taking the upper boundary = -1 and the lower boundary = 3.

∫ ( - 2db/ b^3.

= 2 ∫ ( - db/ b^3.

= 2 [ b^-3 + 2) / -3 + 1.

= 2 | - 1/ 2 db

= 8/9.

Not true.