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## Homework Statement

Use a line integral to find the area of the surface that extends upward from the semicircle ##y=\sqrt{9-x^2}## in the ##xy##-plane to the surface ##z=3x^4y##

## Homework Equations

**Parametric Equation for Circle:**

## x = rcos(t) ##

## y = rsin(t) ##

**Line Integral:**

## \int_c F \cdot dr ##

## The Attempt at a Solution

**(Function):**

## F = z = 3x^4y ##

**(Variables):**

## x = 3cos(t) ##

## y = 3sin(t) ##

## dx = -3sin(t)dt ##

## dy = 3cos(t)dt ##

**(Intervals):**

## -3 ≤ x ≤ 3 ##

## 0 ≤ y ≤ \sqrt{9-x^2} ##

## 0 ≤ t ≤ \pi ## (plugged x-interval into ## x = 3cos(t) ##)

**(Solution):**

## \int_{0}^{\pi} 3(3cos^{4}(t))(3sin(t))dt = 27\int_{0}^{\pi} cos^4(t)sin(t)dt = * ##

## * = 27 [\frac {-1} {5} cos^5(t) |_{0}^{\pi}] ##

## = 27 [ \frac {2} {5}]##

**(Result):**

##= \frac {54} {5}##

**(Questions):**

1. Are there any flaws in this thought process?

2. Since I am switching from substituting x and y with functions of t, do I need some kind of Jacobian in the integral that would take this into account, and affect my final answer?

1. Are there any flaws in this thought process?

2. Since I am switching from substituting x and y with functions of t, do I need some kind of Jacobian in the integral that would take this into account, and affect my final answer?