How To Draw Integral With Maple

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Double Integrals

Of import: This worksheet depends on some programs we have written in Maple. You have to execute these first. Click on the + in the box beneath, then follow the directions you run across at the beginning of the subconscious section.

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Programs for double integrals

Part I, Rectangular Coordinates

Maple can summate double integrals as iterated integrals, either numerically or by repeated use of the Fundamental Theorem of Calculus. The catchy part is providing the appropriate limits of integration, just as it is when doing such integrals past paw. For instance, if R is the region, shown below, betwixt the graphs of and , and , nosotros compute the integral of f over R as as follows.

> int(int((x+y)^three,y = 10^2 .. ten),x = 0 .. 1);

Region between y=x and

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If the function f(x,y) were less simple, for example if , we could compute the integral of f over the region R numerically as follows.

> evalf(Int(Int(10*sin(x+y),y = ten^2 .. x),x = 0 .. ane));

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Observe that the Maple command for integrals has the integration range after the role, and then that the "outer" integration range occurs terminal instead of outset every bit it does when we write the double integral using integral signs .

In this worksheet, our focus will be on setting up double integrals, not evaluating them. To assistance you gain skill in doing this we take devised some new Maple commands, programmed in the subsection above, that will draw the region respective to the integration ranges you specify. For case, the region R appearing above was actually fatigued past the new command dydx(x=0..1, y=x^2..10):, although we haven't displayed the command.

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Notice that in the control dydx(x=0..one, y=x^ii..10): the range for the outer integral (the "dx" integral in this case) is given kickoff, every bit it is when writing integrals, although it appears last in the Maple control above.

Naturally, we programmed a similar control dxdy(y=...,ten=...) for double integrals where the outer integral is with respect to y. If we reverse the order of integration in the integral above, we obtain

Hence, we expect the command dxdy(y=0..1, x=y..sqrt(y)) to produce the same region R. It does, indeed, equally shown below.

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When cartoon regions Maple may choose a viewing window that isn't exactly what we want. Another new control, setwindow , will allow us to specify maximum and minimum values for x and y when using the dxdy and dydx commands. For instance, if you want the viewing window be x=-5..five and y=0..4, you can enter setwindow(-five,5,0,iv): earlier the dxdy or dydx command, using the colon as shown. To return to letting Maple make up one's mind on the viewing window, just enter the command setwindow( ): using the empty parentheses as shown. Y'all should reset the window this way for each new instance or problem.

Let's actually use the dxdy command to see that we get the same region R that we discussed above. We will specify a view window with the setwindow command as well.

> setwindow(-.five,i,-.v,1.five): dxdy(y=0..1, x=y..sqrt(y));

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Every bit you come across, the dydx command volition display the double integral in usual printed form and and then draw the region. It will also try to catch some common errors and tell you lot about them, only it isn't foolproof. If y'all make some unanticipated fault you volition go ane of Maple'southward usual error messages, or some unexpected result. Yous should also notation that if yous switch the upper and lower limits of integration on one of the integral signs, the region will remain the same. Notwithstanding, making this switch changes the sign of the integral.

Here's the game. We give you a region, and y'all take to use the dxdy or dydx commands to specify the limits of integration in a double integral that volition integrate a part F(x,y) over the given region. For some regions, you will have to express the integral over the region as the sum of double integrals over simpler regions. Nosotros accept reduced the size of graphic displays in the worksheet to salve space, and then look those you generate using the dydx and other commands to be larger.

Instance 1. Observe integration limits for a double integral to integrate a part F(x,y) over the region shown beneath.

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Get-go of all, you lot should determine on the order of integration that will make the nearly sense for this region. Then see if you tin can draw the boundaries algebraically. Use the dxdy and/or dydx commands to try out your answers. Apply setwindow( ): to have Maple determine on the view window.

Yous can become hints past clicking on the hidden sections below, but experiment yourself before peeking!!

> setwindow( ):

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Hint 1

Hint 2

Hint 3

Solution

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Example 2. To integrate over the L-shaped region below nosotros might break the region into two rectangular regions and add the integrals over these rectangular regions. For case, the integral of F(x,y) over the region shown below can exist written
+ and we tin can draw the region past the command
display(dxdy(y=0..ii,x=0..i), dxdy(y=2..4,x=0..3)); Notation how the nosotros utilise the display command to combine the results of the two dxdy commands.

How can yous write the integral over the region equally the sum of 2 "dydx" (rather than "dxdy") double integrals? How could yous use the display command with two dydx commands to draw the region?

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Homework Issues, Role I

For each of the post-obit regions and for an arbitrary function F(ten,y), write a double integral of " dxdy" or " dydx" blazon (or both if specified) of F(ten,y) over the region. Use the dxdy and dydx commands to experiment, check your piece of work and produce your final reply. Recollect that if you use the setwindow control, the view window volition remain in upshot until yous release it by using the command: setwindow( ):.

Trouble 1. Give both " dxdy" and "dydx" integrals of F(x,y) over the region:

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Trouble ii. The curved boundary is a parabola. Give both "dxdy" and "dydx" integrals of F(x,y) over the region:

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Trouble 3. The curved office is an arc of a circumvolve. Give both "dxdy" and "dydx" integrals.

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Problem 4. Give both "dxdy" and "dydx" integrals.

Function II, Polar Coordinates

In the subsection at the beginning of this worksheet, we have programmed ii other commands, drdtheta , and dthetadr, that depict regions corresponding to double integrals in polar coordinates. The setwindow control has no effect on these polar coordinate commands. You will find it helpful to know a little more about plotting sure curves in polar coordinates, especially in example 2 beneath. Click the + below for more on polar plotting.

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Polar plotting in brief

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Example 3.

> drdtheta(theta= 0..Pi/ii, r=1..2);

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Example iv.

> drdtheta(theta=0..Pi/2, r =0.. ii*cos(theta));

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Instance 5. We tin can integrate over the region in Example 1 with a "d dr" integral:

> dthetadr(r=1..two, theta=0..Pi/2);

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Instance six. You might have to think a flake about why the region looks the style it does hither:

> dthetadr(r=1..ii, theta=-r/2..r/2);

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Homework Problems, Part Ii

For each of the post-obit regions, write a double integral of "drd " or "d dr" type, or both if specified, for an function F(r, ). Use the drdtheta and dthetadr commands to experiment and check your work.

Trouble 5. Do both ways.

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Problem vi.

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Problem vii.

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Problem 8. Practise some experiments!

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MTH 243 Worksheets written by B. Kaskosz and 50. Pakula, Copyright 1999.

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Source: http://www.math.uri.edu/~bkaskosz/trans1/doubint1.html

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