How do you find the volume of the solid with base region bounded by the curve #9x^2+4y^2=36# if cross sections perpendicular to the #x#-axis are isosceles right triangles with hypotenuse on the base?

Answer 1
The volume of the solid can be found by #V=1/4int_{-2}^2(36-9x^2)dx=24#.
Let us look at some details. If we rewrite #9x^2+4y^2=36# as (by dividing by 36) #x^2/2^2+y^2/3^2=1#, we realize that it is en ellipse that spans from #x=-2# to #x=2#.
By solving for #y#, we have #y=pm1/2sqrt{36-9x^2}# Since the area #A(x)# of the cross-sections can be expressed as
#A(x)=1/2#(base)(height) #=1/2 cdot sqrt{36-9x^2}cdot 1/2sqrt{36-9x^2}# #=1/4(36-9x^2)#
So, the volume of the solid can be found by #V=1/4int_{-2}^2(36-9x^2)dx# by symmetry about the y-axis, #=1/2int_0^2(36-9x^2)dx# #=1/2[36x-3x^3]_0^2=1/2(72-24)=24#
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Answer 2

To find the volume of the solid with base region bounded by the curve (9x^2 + 4y^2 = 36) if cross sections perpendicular to the x-axis are isosceles right triangles with hypotenuse on the base, you can use the method of integration. First, find the equation for the curve in terms of y. Then, integrate the area of a single cross section over the given range of x-values. This involves integrating the area of an isosceles right triangle, which is (\frac{1}{2} \times \text{base} \times \text{height}). The base of each triangle corresponds to the width of the cross section, and the height corresponds to the length of one of the legs. Finally, integrate the expression for the area of each cross section with respect to x over the interval that covers the entire base region. This integral will give you the volume of the solid.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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