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Chapter 17

Chapter 17. Section 17.3 The Evaluation of Double Integrals by Repeated Integrals. y. y. Double Integrals A double integral of a function of 2 variables requires a region in the xy -plane and an area unit . Area units for 2 variables come in two different types. y.

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Chapter 17

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  1. Chapter 17 Section 17.3 The Evaluation of Double Integrals by Repeated Integrals

  2. y y Double Integrals A double integral of a function of 2 variables requires a region in the xy-plane and an area unit . Area units for 2 variables come in two different types. y The Double Integral: Where x x x The area unit that is used determine 2 things that are required to evaluate the integral. The limits that are placed on each integral. The order that you integrate each variable in from the inside integral to the outside. The Area unit for In this case the region is bounded by top and bottom curves and and left and right end points a and b. a b The Area unit for In this case the region is bounded by left and right curves and and top and bottom end points c and d. d c

  3. Example Set up and evaluate double integrals using both types of area units to evaluate the integral to the right.Where is the region bounded by and . Begin by graphing the region . For other integral solve each equation for x. 4 Find projection on the y-axis by plugging in. If then If then Find where curves intersect by solving. and 0 2 0 1st Integral 2 0 1st Integral Treat x as variable. Treat y as variable. Plug in functions for x variable. Plug in functions for y variable. Evaluate like an ordinary integral. Evaluate like an ordinary integral.

  4. The order that you integrate a double integral in will not matter (i.e. give the same answer) as long as you have the correct corresponding limits of integration: To determine which is better to use graph the region and see if it has a single function that forms it top, bottom, right and left boundaries. If a boundary is not a single function the double will need to be split into more than one double integral. y y y x x x Use either: or top, bottom, left and right are all one function Use: top and bottom are different functions Use: left and right are different functions

  5. Example Set up a single double integral and evaluate for the integral to the right where is the region bounded by: , , and . Start by graphing . This region does not always have the same function as its bottom boundary. Sometimes it is a line and some times a parabola. To have one double integral we will need to use . To do this solve to get x as a function of y.

  6. Example Set up double integrals and evaluate them for the integral to the right where is the region bounded by: , , and . To be able to do this in this order we must split this into two regions and which always have the same top and bottom curves. Adding together the values of these two double integrals we get which is the same.

  7. Sometimes reversing the order of integration will give a much easier double integral to evaluate. This means you will need to graph the region from the limits of the integral. Example Reverse the order of integration for the double integral to the right and evaluate it. Begin by graphing the region is bounded by the curves and for x between 0 and 1. Solve each equation for x in order to be able to find the new limits of integration. Let so,

  8. Geometric Interpretation of Integrals If where (i.e. the curve is above the x-axis), then the value of the integral is the area under the curve. Area under curve and above x-axis between a and b: z Volume under the surface and above the region in the xy-plane: For the double integral of a positive surface this is the volume under the surface and above the region in the xy-plane. y x y To find the volume under a surface you need to draw the graph of how the volume you are interested in finding will project into the xy-plane and this becomes your region of integration . x

  9. Example Set up and evaluate a double integral to find the volume of the solid in the 1st octant bounded by the three planes , , and along with the parabolic cylinder and the plane z y Begin by drawing the solid and its projection in the xy-plane. Projection in the xy-plane. y x 2 x

  10. Integrals as Limits A 1 variable integral is the limit of a Riemann Sum. Where is the limit of the and the integral is the limit of the sum of the heights of the rectangles. Area under curve and above x-axis between a and b: b a z For the double integral we cut the region into a grid and look at what happens when we take the limit as both n and m go to infinity, i.e. the mesh of the grid gets smaller and smaller. y x y Volume under the surface and above the region in the xy-plane: x

  11. Area between curves: The area between two curves and can be found by integrating the difference of the functions. z The volume between two surfaces and can be found by the double integral over the projected region in the xy-plane. Volume between the surfaces over : y x Again the region is the projection in the xy-plane of where the top and bottom surface come together.

  12. Example Find the volume of the solid that is bounded on the right by the plane , on the left by the parabolic cylinder , below by the plane , and above by the parabolic cylinder by setting up and evaluating a double integral. First draw the solid along with the projection into the xy-plane. The volume is the double integral: y z 4 3 x y 4 -2 2 x

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