DOUBLE INTEGRAL HELP!!?
I need assistance with converting the following double integral to polar coordinates and evaluating it:
∫(0 to 5/√2) ∫(y to √(25−y^2)) xy dx dy.
Thank you!
1 Answers
The region of integration is x = y to x = √(25−y^2) with y in [0, 5/√2].
==> Inside x^2 + y^2 = 25 in the first quadrant under y = x.
==> Inside r = 5 with θ in [0, π/4] using polar coordinates.
So, ∫(y = 0 to 5/√2) ∫(x = y to √(25 – y^2)) xy dx dy
= ∫(θ = 0 to π/4) ∫(r = 0 to 5) (r cos θ)(r sin θ) * (r dr dθ), via polar coordinates
= ∫(θ = 0 to π/4) sin θ cos θ dθ * ∫(r = 0 to 5) r^3 dr
= (1/2) sin^2(θ) {for θ = 0 to π/4} * (1/4)r^4 {for r = 0 to 5}
= 625/16.
I hope this helps!
Related Questions
Question 1 (Yes/No Worth 1 point): Another expression for "it is raining" is "Está nublado." (Yes/No) Question 3 (Fill-i...
I need assistance with an Algebra 2 problem. The question states that the variable ( r ) varies directly with ( s ) and...
I hope this message finds you well. I have a few questions regarding Frank Garza’s Jockey School. I want to clarify that...
I'm in 12th grade, and I have my physics board exam tomorrow. How can I effectively cover the entire course material in...
A rescue pilot needs to drop a survival kit while flying at an altitude of 2,000.0 meters with a forward velocity of 100...
Hi everyone, I’m trying to make a decision between enrolling my child in a preschool or a Montessori program. He’s curr...
What is the hybridization of the phosphorus atom in PCl3? Additionally, could you please provide an explanation of the s...
I just got hired for a part-time sales job at Home Depot, and I have my orientation tomorrow. I've never had a job where...
What view in a presentation program allows you to display your slides in full-screen mode?
What factors contribute to the observed differences in SAT scores among African American students compared to their Whit...