Let f(x,y)=x2ex2 and let R be the triangle bounded by the lines x=3, x=y/2, and y=x in the xy-plane. (a) Express ∫RfdA as a double integral in two different ways by filling in the values for the integrals below. (For one of these it will be necessary to write the double integral as a sum of two integrals, as indicated; for the other, it can be written as a single integral.)

[tex]\int \int_{R} f(x,y) dA = \int_{0}^{3} \int_{x}^{2x} x^2 e^{x}^{2} dy dx......(1)\\\\\int \int_{R} f(x,y) dA = \int_{0}^{3} \int_{\frac{y}{2}}^{y} x^2 e^{x}^{2} dx dy + \int_{9}^{3} \int_{\frac{y}{2}}^{3} x^2 e^{x}^{2} dx dy..........(2)[/tex]

ashishdwivedilVT ashishdwivedilVT · Answer 2 · 2022-03-26T14:37:40+01:00

By changing the order of integration the integral [tex]\iint_R{}^{}f(x,y)dA =\int_{0}^{3}\int_{x}^{2x} (x^2e^{x^2})dy dx[/tex] split into the sum of two integrals.

Given [tex]f(x,y) = x^{2} e^{x^{2} }[/tex]

R is bounded by the triangle x=3, x=y/2, and y =x in the XY plane.

Let us represent the triangle subtended by x=3, x=y/2, and y =x on coordinate axes.

Find the attached diagram of the triangle bounded by x=3, x=y/2, and

y =x.

[tex]\iint_R{}^{}f(x,y)dA =\int_{0}^{3}\int_{x}^{2x} (x^2e^{x^2})dy dx[/tex]........(1)

By changing the order of integration, (1) becomes

[tex]\iint_R{}^{}f(x,y)dA =\int_{0}^{3}\int_{y/2}^{y} (x^2e^x^)dx dy + \int_{3}^{6}\int_{y/2}^{3} (x^2e^x^)dx dy[/tex]

Thus, By changing the order of integration the integral [tex]\iint_R{}^{}f(x,y)dA =\int_{0}^{3}\int_{x}^{2x} (x^2e^{x^2})dy dx[/tex] split into the sum of two integrals.

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