Evaluate: int_0^9 int_(y/3)^(sqrt(3))(xy^2)dx dy

Loriezon Claveria

Loriezon Claveria

Answered question

2022-07-13

 

Answer & Explanation

Jeffrey Jordon

Jeffrey Jordon

Expert2022-11-07Added 2605 answers

Remove parentheses.

09y33xy2dxdy

Evaluate y33xy2dx.

Since y2 is constant with respect to x, move y2 out of the integral.

09y2y33xdxdy

By the Power Rule, the integral of x with respect to x is 12x2.

09y212x2]y33dy

Simplify the answer.

09y2(32-12(13y)2)dy

Simplify.

093y22-y418dy

Evaluate 093y22-y418dy.

 

Split the single integral into multiple integrals.

093y22dy+09-y418dy

Since 32 is constant with respect to y, move 32 out of the integral.

3209y2dy+09-y418dy

By the Power Rule, the integral of y2 with respect to y is 13y3.

3213y3]90+∫90−y418dy3213y3]09+09-y418dy

Since -1 is constant with respect to y, move -1 out of the integral.

3213y3]09-09y418dy

Since 118 is constant with respect to y, move 118 out of the integral.

3213y3]09-(11809y4dy)

By the Power Rule, the integral of y4 with respect to y is 15y5.

3213y3]09-11815y5]09

Substitute and simplify.

Evaluate 13y3 at 9 and at 0.

32((1393)-1303)-11815y5]09

Evaluate 15y5 at 9 and at 0.

32(1393-1303)-118(1595-1505)

Simplify.

-14585

The result can be shown in multiple forms.

Exact Form:

-14585

Decimal Form:

-291.6

Mixed Number Form:

-291 35


 

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