"Evaluate the integration ∫∫Rr3sin⁡(2θ)drdθ" solved and explained

High School
Geometry
High school geometry
Circles

Chesley

Answered question

2021-08-08

Evaluate the integration

\int \int_{R} r^{3} \sin (2 θ) d r d θ

Answer & Explanation

funblogC

Skilled2021-08-09Added 91 answers

Calculation

2021-12-13

Compute the definite integral:
$\int_{2}^{4} x^{3} \sin (2 Θ) d x$

Factor out constants:
$= \sin (2 Θ) \int_{2}^{4} x^{3} d x$

Apply the fundamental theorem of calculus.
The antiderivative of $x^{3}$ is $\frac{x^{4}}{4}$ :
$= \frac{1}{4} x^{4} \sin (2 Θ)$ right bracketing ${(x = 2)}^{4}$

Evaluate the antiderivative at the limits and subtract.
$\frac{1}{4} x^{4} \sin (2 Θ)$ right bracketing ${(x = 2)}^{4} = \frac{1}{4} 4^{4} \sin (2 Θ) - \frac{1}{4} 2^{4} \sin (2 Θ) = 60 \sin (2 Θ) :$
Answer: $= 60 \sin (2 Θ)$

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