Chapter 5: Q14P (page 268)

Express the integral $I = \int_{0}^{1} d x \int_{0}^{\sqrt{1 - x^{2}}} e^{- x^{2} - y^{2}} d y$ as an integral in polar coordinates $(r, θ)$ and so evaluate it.

Short Answer

Expert verified

The value of Integral is $I = \frac{ττ}{4} (1 - e^{- 1})$

Step by step solution

Given Integral

The given integral is, $I = \int_{0}^{1} dx \int_{0}^{\sqrt{1 - x^{2}}} e^{- x^{2} - y^{2}} dy$

Concept of converting integral to polar coordinates

Use $x = r c o s$ , $r = \sin θ$ and $d A = r d r d θ$ to convert an integral in rectangular coordinates to an integral in polar coordinates.

Evaluate the integral

It is required to express the integral as an integral in polar coordinates $(r, θ)$ and evaluate it.

The area of integration is the quarter-circle in the first quadrant, of unit radius.

Thus, the integral become:

$I = \int_{0}^{1} dx \int_{0}^{\sqrt{1 - x^{2}}} e^{- x^{2} - y^{2}} dy I = \int_{r = 0}^{1} \int_{θ = 0}^{τ τ / 2} e^{- r} rdrdθ I = \frac{ττ}{2} \int_{0}^{1} e^{- r^{2}} d (- r^{2}) (\frac{1}{2}) I = \frac{ττ}{4} (1 - e^{- 1})$

The value of Integral is $I = \frac{ττ}{4} (1 - e^{- 1})$ .

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Express the integral $I = \int_{0}^{1} d x \int_{0}^{\sqrt{1 - x^{2}}} e^{- x^{2} - y^{2}} d y$ as an integral in polar coordinates $(r, θ)$ and so evaluate it.

Short Answer

Step by step solution

Given Integral

Concept of converting integral to polar coordinates

Evaluate the integral

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Express the integralI=∫01dx∫01-x2e-x2-y2dyas an integral in polar coordinates (r,θ) and so evaluate it.

Short Answer

Step by step solution

Given Integral

Concept of converting integral to polar coordinates

Evaluate the integral

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Most popular questions from this chapter

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Express the integral $I = \int_{0}^{1} d x \int_{0}^{\sqrt{1 - x^{2}}} e^{- x^{2} - y^{2}} d y$ as an integral in polar coordinates $(r, θ)$ and so evaluate it.