Calc. Double Integrals: Setting Limits

[mkkrnfoo85]

Here is the question:

Let D be the region bounded by the y-axis and the parabola x = -4y^2 + 3. Compute 'double integral: (x^3)y dxdy'

I'm having a hard time setting limits to the double integral. I got y = [-sqrt(3/4), +sqrt(3/4)] for one of the limits, but I don't know how to set the x limits for integration. Also, can someone show me an easy way to quickly find the limits of integration. Thanks.

 

[whozum]

For double integrals you always want the limits of the outer integral to be constants. Occasionally both integrals will have constant limits.

$$\int \int_D x^3y dx dy$$

The region D is contained by x = 0 and x = 3-4y^2. Since your outer integral is dy, your y limits will be constant, and x limits will be variable. Your limits alon the x-axis are given by x = 0, and x = 3-4y^2, and:

$$\int \int_0^{3-4y^2} x^3y dx dy$$

Plot the graph of your domain to determine the range of y. You will see that the maximum y value occurs on the y axis, at x = 0.

Solving 0 = 3-4y^2, you find y = +/- sqrt(3/4).


The easiest way to find limits of integration is to sketch the region of integration.

The final integral becomes $$\int_{-\sqrt{3/4}}^{\sqrt{3/4}} \int_0^{3-4y^2} x^3y dx dy$$

 

[thepatientmental]

The limits of integration for a double integral can be found by graphing the region D and identifying the boundaries. In this case, D is bounded by the y-axis and the parabola x = -4y^2 + 3. To find the x limits, we can solve for x in terms of y by rearranging the equation of the parabola: x = (-4y^2 + 3). This gives us the lower and upper limits of x as -4y^2 + 3 and 0, respectively. Therefore, the double integral can be written as:

∫∫(x^3)y dxdy = ∫[0, √(3/4)]∫[-4y^2 + 3, 0](x^3)y dxdy

To quickly find the limits of integration, you can also use the method of slicing, where you divide the region D into vertical or horizontal strips and integrate over each strip separately. In this case, you can divide D into vertical strips and integrate over each strip from y = -√(3/4) to y = √(3/4). This will give you the same limits of integration as found above. I hope this helps!