Finding the Area of a Shaded Region

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In summary, the total area of the shaded region can be found by calculating the integral: ##\displaystyle \int_{\pi/4}^{3 \pi/4} 4\sqrt{2} - 4\csc{\theta}\cot{\theta} \ d\theta##, where the first term represents the area of the rectangle and the second term subtracts the area under the curve. By subtracting the second term, the area under the curve is added to the rectangle's area. This integral can also be solved by separating it into two parts. Alternatively, it could be solved using a double integral, but this may not be required for this problem.
  • #1
Youngster
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Homework Statement



Find the total area of the shaded region

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Homework Equations



The Fundamental Theorem of Calculus: [itex]\int[/itex][itex]^{b}_{a}[/itex] f(x) dx = F(b) - F(a)

The Attempt at a Solution



I don't seem to have a clue about how to approach this one. Though for previous total area problems, I had to take the absolute value of specific intervals and then add them to obtain the area, or subtract the area under the curve by forming a rectangle using the boundaries given, and subtracting the area of the curve from the area of the rectangle.

Technically I could make a rectangle in this problem, but do I have to? I don't really have any examples in my textbook that point me towards any strategy here.
 
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  • #2
Technically the area would just be the area of the rectangle because the area under the x-axis has to equal the area from ∏/4 to ∏/2.

I think that would be right?
 
  • #3
Ah you're right! That bottom portion should fit into the top portion to make a good rectangle :O

Is there some way to calculate that though?
 
  • #4
Think basic ;)

Area of a rectangle = Base x Height
 
  • #5
Sorry, haha. I mean, using antiderivatives considering this chapter focuses on using those to calculate areas. I'm aware that it's much simpler to just form a rectangle of [itex]\frac{\pi}{2}[/itex] base and 4[itex]\sqrt{2}[/itex] height, and multiply those values to get the area, but I'm curious as to how one obtains it through the use of antiderivatives.
 
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  • #6
Youngster said:
Sorry, haha. I mean, using antiderivatives considering this chapter focuses on using those to calculate areas. I'm aware that it's much simpler to just form a rectangle of [itex]\frac{∏}{2}[/itex] base and 4[itex]\sqrt{2}[/itex] height, and multiply those values to get the area, but I'm curious as to how one obtains it through the use of antiderivatives.

You would calculate the integral: ##\displaystyle \int_{\pi/4}^{3 \pi/4} 4\sqrt{2} - 4\csc{\theta}\cot{\theta} \ d\theta##

This is because the ##4\sqrt{2}## term calculates the area of the box. By subtracting the second part, you take away from the first part of the box, and then you ADD the second part of the integral of the curve. Normally the area is negative for the part of the curve below the x-axis, but when you subtract it becomes positive.
 
  • #7
Ah alright! That worked for me.

I was on the right track before, but I went ahead and separated the integral into two parts so that I'd have one integral from [itex]\frac{\pi}{4}[/itex] to [itex]\frac{\pi}{2}[/itex] and another from [itex]\frac{\pi}{2}[/itex] to [itex]\frac{3\pi}{4}[/itex]. I see how that works now though.

Thank you for the help!
 
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  • #8
Well, you could solve this using double integral, which i doubt you've covered yet in your syllabus. But i don't think that's what you're being required to do in this problem. You should just solve it as suggested above.
 

1. How do you find the area of a shaded region?

To find the area of a shaded region, you need to first identify the shape of the shaded region. Then, you can use the appropriate formula to calculate the area of that shape. Finally, you can subtract any non-shaded regions from the total area to get the area of the shaded region.

2. What is the formula for finding the area of a shaded region?

The formula for finding the area of a shaded region depends on the shape of the shaded region. Some common formulas include:
- Rectangle: length x width
- Triangle: 1/2 x base x height
- Circle: π x radius^2

3. Can you provide an example of finding the area of a shaded region?

Sure! Let's say we have a rectangle with a length of 10 inches and a width of 5 inches. The total area of the rectangle would be 10 x 5 = 50 square inches. If there is a shaded region taking up 20 square inches, the area of the shaded region would be 50 - 20 = 30 square inches.

4. What if the shaded region is made up of multiple shapes?

If the shaded region is made up of multiple shapes, you can find the area of each shape individually using the appropriate formula. Then, you can add the areas together to get the total area of the shaded region.

5. How do you know which regions to include in the calculation of the shaded region's area?

To calculate the area of a shaded region, you should only include the regions that are fully shaded. If there are any partial or overlapping shaded regions, you may need to adjust your calculation accordingly.

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