calc_tutorial_15_2_015

 
Problem Statement

{5,7,11,13}
2*7
round(7*11/60,2)

Calculate the integral of f(x, y) = 7x over the region D bounded above by y = x(2 − x) and below by x = y(2 − y). Hint: Apply the quadratic formula to the lower boundary curve to solve for y as a function of x.

 
Step 1

Question Sequence

Question 1

The first step is to understand the region D.

The upper bound, y = x(2 − x), is a that opens with x-intercepts at x = 0 and x = 2.

The lower bound, x = y(2 − y), is a that opens with y-intercepts at y = 0 and y = 2.

With this information, graph the region.

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 2

Consider the following figure.

We could treat the region D as either vertically or horizontally simple, but since the integrand, x, is without the variable y, it will be easier to consider D as vertically simple.

In order to do so, we need the bounding functions in terms of x. Hence we must solve the lower bound, x = y(2 − y), for y.

Use the quadratic equation to solve for y in terms of x.

x = y(2 − y)

y2 − 2y + x = 0

y =

A.
B.
C.
D.

The two solutions correspond to the upper and the lower branch of the parabola. The lower bound of D is only one of these branches.

Find the proper branch that bounds D from below. (Hint: This branch goes to −infinity.)

y =

A.
B.

State the inequalities that define region D

0 ≤ x ≤

and

1 √(1-x) ≤ y ≤ x(2 − x)

Correct.
Incorrect.

 
Step 3

Question Sequence

Question 3

Set up the integral considering the region D as vertically simple.

,

where b =

A.
B.
C.
D.
Correct.
Incorrect.

 
Step 4

Question Sequence

Question 4

Integrate. (Hint: For the x√(1 − x) term, integrate using u-substitution.) Round your answer to two decimal places.

=

Correct.
Incorrect.