Evaluating the integral of absolute values

In summary, the integral ∫(0 to 3pi/2) -7|sinx|dx requires breaking the interval into cases where x ≥ 0 and x < 0. This is because the absolute value of sinx changes depending on the value of x. By using the appropriate substitution for each case, the correct solution can be found.
  • #1
doctordiddy
54
0

Homework Statement



∫(0 to 3pi/2) -7|sinx|dx

Homework Equations





The Attempt at a Solution



I am not sure how to treat it as it has an absolute value

i assumed that you could remove the -7 to get

-7∫|sinx| dx

then integrate sinx into -cosx but since there is absolute value i tried to change -cosx to cosx which ended up as

-7cosx

and then doing -7cos(3pi/2)-(-7cos(0))

but this was incorrect. Can anyone give me any hints?

thanks
 
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  • #2
doctordiddy said:

Homework Statement



∫(0 to 3pi/2) -7|sinx|dx

Homework Equations





The Attempt at a Solution



I am not sure how to treat it as it has an absolute value

i assumed that you could remove the -7 to get

-7∫|sinx| dx

then integrate sinx into -cosx but since there is absolute value i tried to change -cosx to cosx which ended up as

-7cosx

and then doing -7cos(3pi/2)-(-7cos(0))

but this was incorrect. Can anyone give me any hints?

thanks

Usually for absolute values you have to break things into cases depending on what values of x you're integrating over.

Notice that from 0 to 3π/2, x≥0? If x≥0, then |sinx| = sinx.

Consider integrating from x=-2 to x=-1, x<0. If x<0, then |sinx| = -sinx.
 
  • #3
doctordiddy said:

Homework Statement



∫(0 to 3pi/2) -7|sinx|dx

Homework Equations





The Attempt at a Solution



I am not sure how to treat it as it has an absolute value

i assumed that you could remove the -7 to get

-7∫|sinx| dx

then integrate sinx into -cosx but since there is absolute value i tried to change -cosx to cosx which ended up as

-7cosx

and then doing -7cos(3pi/2)-(-7cos(0))

but this was incorrect. Can anyone give me any hints?

thanks
Sketch the graph of y = sinx from 0 to 3pi/2. From this, can you see what g = |sinx| would look like? Do you then see why integrating sinx to simply -cosx is wrong?
 
  • #4
Zondrina said:
Usually for absolute values you have to break things into cases depending on what values of x you're integrating over.

Notice that from 0 to 3π/2, x≥0? If x≥0, then |sinx| = sinx.
This is NOT true. If 0 ≤ x ≤ π, then |sinx| = sinx, but for π ≤ x ≤ 2π, sin(x) ≤ 0.
Zondrina said:
Consider integrating from x=-2 to x=-1, x<0. If x<0, then |sinx| = -sinx.

This is not true, either. There are infinitely many intervals for which x < 0 but sin(x) ≥ 0.
 
  • #6
It is still not right. Can you use |sinx| = sinx if sinx ≥ 0 and |sinx| = -sinx if sinx < 0? This is key to solving the problem.
 

FAQ: Evaluating the integral of absolute values

1. What is the definition of an integral of absolute values?

The integral of absolute values refers to the process of finding the area under the curve of an absolute value function. It is a mathematical concept used in calculus to determine the total change or accumulation of a quantity over an interval.

2. How do you evaluate the integral of absolute values?

To evaluate the integral of absolute values, you must first determine the limits of integration, which are the values that define the beginning and end points of the interval. Then, you can use various integration techniques, such as substitution or integration by parts, to find the antiderivative of the absolute value function. Finally, you can plug in the limits of integration to calculate the definite integral.

3. What are the properties of integrals of absolute values?

Some important properties of integrals of absolute values include linearity, where the integral of a sum is equal to the sum of the integrals, and the fact that the integral of an absolute value function is always positive. Additionally, the integral of an absolute value function can be split into smaller intervals and added together to find the total area.

4. How do you handle absolute values in integration?

When integrating a function that contains absolute values, you can rewrite the absolute value as a piecewise function with different expressions for the function depending on whether the input is positive or negative. Then, you can integrate each piece separately and combine the results to find the final integral.

5. What are some real-world applications of evaluating integrals of absolute values?

Evaluating integrals of absolute values has many practical applications in fields such as physics, engineering, and economics. For example, it can be used to calculate the displacement of an object moving at a varying velocity, determine the total cost of a changing product, or find the net force acting on an object with varying acceleration.

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