# Calculus 3 : Double Integrals

## Example Questions

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### Example Question #11 : Double Integration Over General Regions

Let  be any continuous, one-to-one function over the interval , with on  and . Select all integrals which correctly define the indicated area under the curve.

3 and 4

1 and 4

1 and 3

1

1 3 and 4

1 and 3

Explanation:

No explanation necessary. This is the familiar integral for computing the area under the curve  over

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This integral does not give the area, in fact, it will result in a function of  which certainly cannot be interpreted as the area of corresponding to a specific interval

Now carry out the integration with respect to  we obtain,

Certainly this is not the area of the area under . One obvious problem is that the integral gave back a function of  and no real number.

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This equation will reduce down to the first option, option , after we carry out the integration with respect to .

Therefore, option  is a valid representation for the area.

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The problem with this choice can be seen immediately. Clearly the first integration in will simply be the area under the curve on , but the second will now multiply the area by

could only be true if . But because none of the listed characteristics of our function imply this, we cannot assume it. In fact, the problem stated that  is any function that satisfies the stated conditions. Certainly  is not true for just any function.

### Example Question #12 : Double Integration Over General Regions

Evaluate the following iterated integrals:

Explanation:

When evaluating double integrals, work from the inside out: that is, evaluate the integrand with respect to the first variable listed by the differential operators, and then evaluate the result with respect to the second variable listed by the differential operators.

Here, we have the order of integration specified by ; hence, we evaluate the double integrals with respect to  first, and then integrate the result with respect to , as shown:

### Example Question #21 : Double Integration Over General Regions

Calculate the following Integral.

Explanation:

Lets deal with the inner integral first.

Now we evaluate this expression in the outer integral.

### Example Question #22 : Double Integration Over General Regions

Calculate the following Integral.

Explanation:

Lets deal with the inner integral first.

Now we evaluate this expression in the outer integral.

### Example Question #681 : Multiple Integration

Evaluate , where  is the trapezoidal region with vertices given by , and ,

using the transformation , and .

Explanation:

The first thing we have to do is figure out the general equations for the lines that create the trapezoid.

Now we have the general equations for out trapezoid, now we need to plug in our transformations into these equations.

So our region is a rectangle given by

Next we need to calculate the Jacobian.

Now we can put the integral together.

### Example Question #1 : Surface Area

Find the surface area of the part of the plane  in the first octant.

Explanation:

Lets recall the equation of surface area.

Now we need to find all the neccessary equations to be able to evaluate the integral.

We will plug in , into the plane equation in order to get a line that intersects with the z axis.

Now we are going to set , in the previous equation and solve for .

We now have all the bounds for our double integral

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