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Use a CAS double-integral evaluator to find the integrals. Then reverse the order of integration and evaluate, again with a CAS.$$\int_{0}^{1} \int_{2 y}^{4} e^{x^{2}} d x d y$$

$$1.1494 \times 10^{6}$$

Calculus 3

Chapter 15

Multiple Integrals

Section 2

Double Integrals over General Regions

Johns Hopkins University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:18

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. The area above the x-axis adds to the total.

26:18

In mathematics, a double integral is an integral where the integrand is a function of two variables, and the integral is taken over some region in the Euclidean plane.

03:17

Use a CAS double-integral …

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the gunman double integral is 0 to 2 0 to 4 minus y square you to the poetics way bx doing Onda. We have to use a graphing calculator to estimate the value of this integral. So we'll use a graphing calculator on the estimated value for this Integral by our calculator comes out to be as 20.56 Faith. Now we have to write this integral by reversing the order of integration. So for that, we know that these are the limits for X, so X is equal to zero. Annexes equals to four minus y square X. Now these other Mitch for why that win survive is equal to zero y equals Toto. I'll be graphing these equation on a graph paper to sketch the reason off integration. So by sketching region of integration, we can write this integration by reversing the order of integration. So let's hear a good our favor. No. So this go represents the parable of which is exit equals two for minus y square. On this line is like worlds to to hair vice equals 20 on Hair X is equal to zero on the reason off integration is this region. No, we have to write the interval by reversing the order of integration. So you get the Dublin trickle into the Power X Y divide the X. We have just reverse the order of integration. Now the auto writer limits for that. We have to take a vertical strip so hair why is equal to zero? So the limit would the lower limit would be zero on the upper limit is here. Y is equal. It's Do we know that the equation of this line is X is equal to four minus y squared or we can write Vice squared is equal to four minus X, so why comes out to be as under wrote four minus six. So apart limit would be four minus x. Now we're right. The limits for a D. X so x varies from this line to this lane on The question of this line is X is equal to zero to the lower role limit is zero Onda Parliament would be X is equal to four because they could, you know this line is executed for apart limited For now again, we have really a graphing calculator on a year to find the will of the symmetrical, which comes out to be as 20.6 35 6 Fife. So finally we can conclude that the integral zero toto 0 to 4 minus y square you to the politics way DX dy y is equal to irritable integral 0 to 40 toe under mode four minus X to the Power X Y divide DX, and it comes out to be as equal to 20.565 So this is the final reserved.

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