Let's break down and solve the two parts one by one: --- ## **Part 1: Translate the integral into rectangular coordinates** Given: \[ \int_{0}^{2\pi}\int_{0}^{3}\int_{0}^{4} r \cos(θ) \, dr \, dθ \, dz \] This is an integral in **cylindrical coordinates**. The variables are: - \( r \) = radius (distance from \( z \)-axis) - \( θ \) = angle (from \( x \)-axis) - \( z \) = height ### **Step 1: Identify the bounds and function** - \( r \) from 0 to 3 - \( θ \) from 0 to \( 2\pi \) - \( z \) from 0 to 4 - The function is \( r \cos θ \) ### **Step 2: Convert function to rectangular coordinates** Recall the conversions: - \( x = r \cos θ \) - \( y = r \sin θ \) - \( r = \sqrt{x^2 + y^2} \) - \( θ = \tan^{-1}(y/x) \) - \( z = z \) So, \( r \cos θ = x \). ### **Step 3: Convert the bounds** - \( r \) from 0 to 3: This describes a circle of radius 3 in the \( xy \)-plane. - \( θ \) from 0 to \( 2\pi \): Full circle. - \( z \) from 0 to 4: Standard. So, the region in rectangular coordinates is: - \( x^2 + y^2 \leq 9 \) - \( 0 \leq z \leq 4 \) ### **Step 4: The integral in rectangular coordinates** The volume element in cylindrical is \( r dr dθ dz \), and in rectangular is \( dx\,dy\,dz \). So, the integral becomes: \[ \int_{z=0}^{4} \int_{x^2+y^2 \leq 9} x \, dx \, dy \, dz \] --- ## **Part 2: Translate the point \((-7, 7, 4)\) from rectangular to cylindrical coordinates** Given: - \( x = -7 \) - \( y = 7 \) - \( z = 4 \) ### **Step 1: Convert to cylindrical coordinates** Formulas: - \( r = \sqrt{x^2 + y^2} \) - \( θ = \tan^{-1}(y/x) \) - \( z = z \) Calculate: - \( r = \sqrt{(-7)^2 + (7)^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2} \) - \( θ = \tan^{-1}\left(\frac{7}{-7}\right) = \tan^{-1}(-1) \) Note: \((x, y) = (-7, 7)\) is in the **second quadrant**. For \(\tan^{-1}(-1)\), the principal value is \(-\frac{\pi}{4}\), but in the second quadrant, add \(\pi\): \[ θ = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \] So, the cylindrical coordinates are: \[ (r, θ, z) = (7\sqrt{2}, \frac{3\pi}{4}, 4) \] --- ## **Final Answers** ### **1. Integral in rectangular coordinates:** \[ \boxed{ \int_{z=0}^{4} \int_{x^2+y^2 \leq 9} x \, dx \, dy \, dz } \] ### **2. Point in cylindrical coordinates:** \[ \boxed{ (7\sqrt{2},\ \frac{3\pi}{4},\ 4) } \]