Double Integral with a trig function

In summary, the conversation discusses integrating cos(x + y) with upper limits of pi/2 and lower limits of 0 for both integrals. Mark44 confirms that the answer is 0 and shares a different method using the identity cos(x + y) = cos(x)cos(y) - sin(x)sin(y). Another method is mentioned, where you directly integrate cos(x + y) and evaluate it at pi/2 and 0. The summary concludes by mentioning the final steps of integrating cos(y) - sin(y) and evaluating the antiderivative at pi/2 and 0.
  • #1
eddysd
39
0
just wondering if i can still do this, attempted the following:

ʃʃ cos(x+y)dxdy with upper limits of pi/2 and lower limits of 0 for both integrals

My answer came out as 0.

Can anyone confirm this?
 
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  • #2
eddysd said:
just wondering if i can still do this, attempted the following:

ʃʃ cos(x+y)dxdy with upper limits of pi/2 and lower limits of 0 for both integrals

My answer came out as 0.

Can anyone confirm this?

That's what I get.
 
  • #3
@Mark44 Thankyou, also what method did you use, I used the identity that
cos(x+y)=cos(x)cos(y)-sin(x)sin(y)

Is there another way of doing it?
 
  • #4
Yes. You can integrate cos(x + y) directly. When you integrate cos(x + y) with respect to x (treating y as a constant), you get sin(x + y). Evaluating this at pi/2 and 0 gives sin(pi/2 + y) - sin(y). The sin(pi/2 + y) term can be rewritten as cos(y) using an identity.

Finally, integrate cos(y) - sin(y) with respect to y, and evaluate the antiderivative at pi/2 and 0.
 

What is a double integral with a trig function?

A double integral with a trig function is a type of integral in which both the inner and outer integrals contain trigonometric functions. It is used to calculate the area under a curved surface or volume under a curved solid in three-dimensional space.

How do I set up a double integral with a trig function?

To set up a double integral with a trig function, you will need to determine the limits of integration for both the inner and outer integrals. This can be done by sketching the region of integration and identifying the curves or surfaces that define the boundaries.

What are the steps for evaluating a double integral with a trig function?

The steps for evaluating a double integral with a trig function are:

  1. Set up the integral by determining the limits of integration for both the inner and outer integrals.
  2. Simplify the integrand by using trigonometric identities or substitution.
  3. Evaluate the inner integral first, treating the outer limits as constants.
  4. Evaluate the outer integral using the result from the inner integral.

What are some common trig identities used in double integrals?

Some common trig identities used in double integrals include:

  • sin^2(x) + cos^2(x) = 1
  • tan(x) = sin(x)/cos(x)
  • sec^2(x) = 1 + tan^2(x)
  • csc^2(x) = 1 + cot^2(x)

How do I know when to use a double integral with a trig function?

A double integral with a trig function is typically used when the region of integration or the integrand involves trigonometric functions. It is also useful for calculating areas or volumes of curved surfaces or solids.

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