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MA 227 Standard C07

Double Integrals
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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)