What do these double integrals represent?

Question

Question

I have three double integrals:

(A) $$\int_{0}^{1}\int_{x^2}^1 dydx$$

(B) $$\int_{0}^{1}\int_{x}^{2x} x^2 dydx$$

(C) $$\int_{0}^{1}\int_{-y}^{y} dxdy$$

These need to be matched to the appropriate statements:

(a) The area of the triangle in the $xy$-plane corresponds to
(b) The area of a region in the $xy$-plane bounded on a side by a parabola corresponds to
(c) The volume under the surface $z=x^2$ above a triangle in the $xy$ plane corresponds to
(d) The volume under the plane $z=1$ above a triangle in the $xy$ plane corresponds to

My Attempt

So far I have that

(a) C
(b) A
(c) B
(d) C

However I am confused if there can be multiple answers. For example, is it possible that (a) is B and C, since both regions of integration are triangles in the $xy$ plane?

These are my sketches of the regions of integration:

Answer 1

Yes, your answers seem to be correct. Here is an explanation -

For the first integral, for any x the range in y is from $x^2$ to 1. Inside integral, we only have dy and dx and thus this gives the area of the region shown by you in the first figure.

For the second integral, we can follow the same idea. The figure which you drew is also correct. Thus the value of the integral is the volume bounded by this triangle and z=$x^2$.

Following a similar idea, it can be shown that your answer for the third integral is also correct.

Answer 2

B: $$\int_{0}^{1}\int_{x}^{2x} x^2 dydx$$

C: $$\int_{0}^{1}\int_{-y}^{y} dxdy$$

(a) The area of the triangle in the $xy$-plane corresponds

(c) The volume under the surface $z=x^2$ above a triangle in the $xy$ plane corresponds to

is it possible that statement (a) corresponds to integrals B and C, since both regions of integration are triangles in the $xy$ plane?

Let $R$ be a planar region. The integral $$\iint_R \:\mathrm dx\,\mathrm dy$$ gives the area of $R,$ while the integral $$\iint_R f(x,y)\:\mathrm dx\,\mathrm dy$$ gives the signed volume between $R$ and the surface $z=f(x,y).$

Thus, since both integrals B and C have a triangular integration domain in the $x$-$y$ plane,

• statement (a) corresponds only to integral C, and
• statement (c) only to integral B.

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