# MA 227 Standard C07

Double Integrals

At the end of the course, each student should be able to…

• C07: DoubleInt. Compute and apply double integrals.

## C07. Double Integrals

• Just as a single integral measures the area under a curve, a double integral measures the volume under a surface.
• Let $$\iint_R f(x,y)\,dA$$ be the volume of the solid above the region $$R$$ in the $$xy$$ plane and below the surface $$z=f(x,y)$$, minus the volume of the solid below the region $$R$$ in the $$xy$$ plane and above the surface $$z=f(x,y)$$. This volume above minus volume below is called the net volume.
• It follows that $$\iint_R f(x,y)\,dA=\int_a^b A(x)\,dx$$ where $$A(x)$$ is the net area of the cross-section at each $$x$$-value and $$a,b$$ are the leftmost and rightmost $$x$$-values of the region. Similarly, $$\iint_R f(x,y)\,dA=\int_c^d A(y)\,dy$$ for the bottommost/topmost $$y$$-values $$c,d$$.
• Also, $$A(x)=\int_{g_1(x)}^{g_2(x)}f(x,y)\,dy$$ where $$g_1(x)$$ is the bottom curve of the region and $$g_2(x)$$ is the top curve of the region. Similarly, $$A(y)=\int_{g_1(y)}^{g_2(y)}f(x,y)\,dx$$ for the left/right curves $$g_1(y),g_2(y)$$.
• Thus $$\iint_R f(x,y)\,dA= \int_a^b[\int_{g_1(x)}^{g_2(x)}f(x,y)\,dy]\,dx$$ and $$\iint_R f(x,y)\,dA= \int_c^d [\int_{g_1(y)}^{g_2(y)}f(x,y)\,dx]\,dy$$. These nested definite integrals are called iterated integrals.
• Several properties of double integrals are easily proven.
• $$\iint_R cf\,dA=c\iint_R f\,dA$$.
• $$\iint_R (f\pm g)\,dA= \iint_R f\,dA \pm \iint_R g\,dA$$.
• $$f\leq g$$ implies $$\iint_R f\,dA\leq\iint_R g\,dA$$.
• $$\iint_{R+S} f\,dA = \iint_R f\,dA + \iint_S f\,dA$$.
• The order of integration may be reversed by reinterpreting the region of integration.
• When $$R=[a,b]\times[c,d]$$ is a rectangle, $$\iint_R f(x,y)\,dA= \int_a^b\int_c^d f(x,y)\,dy\,dx= \int_c^d\int_a^b f(x,y)\,dx\,dy$$.
• Otherwise, the bounds may only be switched by sketching and reinterpreting the region of integration, considering top/bottom curves rather than left/right curves (or vice versa).
• Double integrals may be applied to find areas and average values.
• The area of $$R$$ is given by $$\iint_R 1\,dA$$.
• The average value of $$f$$ over $$R$$ is given by $$\frac{1}{Area(R)}\iint_R f\,dA$$.

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 14.1 (exercises 1-30)
• 14.2 (exercises 19-28, 33-54)
• 14.3 (exercises 1-12, 19-22)