Worksheet 3.1: Introduction to Double Integrals

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1 Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals Prerequisites In order to learn the new skills and ideas presented in this worksheet, you must: Be able to integrate functions of a single variable. Goals In this worksheet, you will: Evaluate double integrals on rectangular and general domains. Change the order of integration when appropriate to evaluate double integrals.

2 Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals Warm-Up Consider the diagram below: y = x y = x A. Sketch a grid in the region using lines parallel to the x-axis and lines parallel to the y-axis. - B. Choose one of the rectangles in your grid. Use dx to lable the length of a side parallel to the x-axis, and use dy to label the length of the side parallel to the y-axis. Then, if da is the area of the rectangle: da = C. What is the equation of the curve that forms the bottom boundary of the region? What is the equation of the curve that forms the top boundary of the region above?. Find the the x- and y-coordinates of the two points of intersection of the curves bounding.

3 Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals Integrating Over Rectangular omains. (a) Sketch the rectangle x 5, y, and label the curves that make up the boundary. This will be the region of integration for the next few integrals. Pay attention to the limits of integration of the integral, and the boundary curves of the region (b) On your sketch, draw a grid in the region (the rectangle) using lines parallel to the coordinate axes. Pick one of the small rectangles in your grid, and label a side parallel to the x-axis as dx, and a side parallel to the y-axis as dy. What is the area da of the small rectangle, in terms of dx and dy? da is called the area element. da = (c) Evaluate the following integral using the order of integration da = dx dy. First, evaluate the inside integral with respect to x, treating y as a constant. Then evaluate what s left with respect to y: ˆ ˆ 5 ˆ ( ˆ 5 ) 4xy da = 4xy dx dy = 4xy dx dy (d) Now evaluate the same integral using the order of integration da = dy dx (evaluate inside integral with respect to y treating x constant, then evaluate what s left with respect to x) and compare the result with your answer from #. (They should be the same.) ˆ 5 ˆ ˆ 5 ( ˆ ) 4xy da = 4xy dy dx = 4xy dy dx

4 Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals ouble Integrals Over General Regions. Suppose we want to integrate f (x, y) = x over the triangular domain pictured below, using the order of integration da = dy dx. (a) First, find the limits of integration with respect to y (since it is the inside integral). Think about moving through the domain vertically, from bottom to top. raw an arrow parallel to the y-axis, starting below the region and ending above the region : y = y = - x/ + x = The lower limit of integration with respect to y is the equation of the line where the arrow enters at the bottom of. The upper limit of integration with respect to y is the equation of the line where the arrow leaves at the top of. Lower limit with respect to y (arrow enters ) is: y = Upper limit with respect to y (arrow leaves ) is: y = continued

5 Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals 4 (b) Now, find the limits with respect to x. Since this is the outside integral, these limits must be constant! To find the limits with respect to x, project the region onto the x-axis, and find the x-values covered by the region: y = y = - x/ + x = The lower limit of integration with respect to x is the smallest x-value covered by the region. The upper limit of integration with respect to x is the largest x-value covered by the region. Lower limit with respect to x (smallest x-value) is: x = Upper limit with respect to x (largest x-value) is: x = (c) Using the limits of integration found in parts (a) and (b), set up and evaluate the integral: x da = ˆ x= x= ˆ y= y= x dy dx =

6 Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals 5 (d) You can also use the other order of integration, da = dx dy. The inside integral is evaluated with respect to x, so draw the arrow parallel to the x-axis, from left to right. To find the limits with respect to y (which will be constant), project the region onto the y-axis. y = y = - x/ + x = The lower limit of integration with respect to x is the equation of the line where the arrow enters at the left of. You will need to solve for x! The upper limit of integration with respect to x is the equation of the line where the arrow leaves at the right of. Lower limit with respect to x (arrow enters ) is: x = Upper limit with respect to x (arrow leaves ) is: x = The lower limit of integration with respect to y is the smallest y-value covered by the region. The upper limit of integration with respect to y is the largest y-value covered by the region. Lower limit with respect to y (smallest y-value) is: y = Upper limit with respect to y (largest y-value) is: y = continued

7 Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals 6 Using the limits of integration found in part (d), set up and evaluate the double integral. x da = ˆ y= y= ˆ x= x= x dx dy = (e) You should get the same answer for the two integrals in parts (c) and (d) you are integrating the same function (f (x, y) = x) over the same region (the triangle ). Was it easier to integrate using the order of integration da = dy dx from part (c), or the order da = dx dy from part (d)? Which one? Wny? (f) Conclusions: i. The outer limits of integration will always be constant. Always True Sometimes True Never True ii. To change the order of integration, just swap the limits from the first integral to get the limits of the second integral. Always True Sometimes True Never True iii. The value of the integral is the same, regardless of which order of integration you use. Always True Sometimes True Never True

8 Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals 7 ouble Integrals and Area. (a) Consider the integral: da = da = = = = ˆ 4 ˆ ˆ 4 ˆ 4 = ˆ 4 y dx dy dx dx dx i. Using the limits of integration, draw the rectangular domain of this integral ii. What does the integral in (a) represent with respect to the domain you drew in (b)?

9 Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals 8 (b) Consider the integral: da = da ˆ ˆ y = ˆ dx dy = x y dy ˆ = y dy = y / = / i. Using the limits of integration, draw the triangular domain of this integral ii. What does the integral in (a) represent with respect to the domain you drew in (b)? (c) In general, what is represented by the integral: da = da?

10 Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals 9 Changing Order of Integration 4. Sometimes, it may be useful (or necessary!) to reverse the order of integration for a given integral. Consider the integral: ˆ x= ˆ y= e y da = e y dy dx. x= y=x/ (a) Sketch the domain of integration using the limits of integration from this integral. (b) Using your sketch from part (a), find the limits for the order of integration da = dx dy. Use these limits to set up and evaluate the integral with the order of integration reveresed. (c) What do you gain by reversing the order of integration in this case?

11 Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals Regions Requiring More Than One ouble Integral 5. Go back to the region pictured in the Warm-up. (a) Set up the double integral f (x, y) da, using the order of integration da = dy dx. (You don t know the function f (x, y), so you cannot evalute.) (b) Now, set up the double integral f (x, y) da, using the order of integration da = dx dy. How many double integrals do you need for this order of integration? Why?

12 Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals Summary ouble integral of f (x, y) over a planar region : f (x, y) da da is called the area element. It measures the area of an infinitesimal rectangle in the plane. In Cartesian coordinates, da = dx dy = dy dx. is the region of integration in the xy-plane. The boundary curves of determine the limits of integration. If f (x, y) is continuous, and the boundary curves of are continuous, the double integral can be evaluated as two single integrals. The inside integral is evaluated with respect to the indicated variable using techniques from Calc I/II, while holding the other variable constant (compare to partial derivatives!). The limits of integration of the outside integral are always constant. Using the order of integration da = dy dx, with g (x) y g (x) and a x b : f (x, y) da = ˆ b ˆ g (x) f (x, y) dy dx = ˆ b a g (x) a g (x) ( ˆ ) g (x) f (x, y) dy dx Using the order of integration da = dx dy, with h (y) x h (y) and c y d : f (x, y) da = ˆ d ˆ h (y) f (x, y) dx dy = ˆ d c h (y) c h (y) ( ˆ ) h (y) f (x, y) dx dy If f (x, y) is continuous, and the boundary curves of are continuous, the order of integration does not matter: f (x, y) da = ˆ b ˆ g (x) a g (x) f (x, y) dy dx = ˆ d ˆ h (y) c h (y) f (x, y) dx dy

Worksheet 3.2: Double Integrals in Polar Coordinates

Worksheet 3.2: Double Integrals in Polar Coordinates Boise State Math 75 (Ultman) Worksheet 3.: ouble Integrals in Polar Coordinates From the Toolbox (what you need from previous classes): Trig/Calc II: Convert equations in x and y into r and θ, using the