Find the area of the region between the curves y=x^4 and y=x^2
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First find where they intersect.
x^4 = x^2
x^4 - x^2 = 0
x²(x² - 1) = 0
x = -1, x = 0, x = 1
∫ (x^2 - x^4) dx
= [ (1/3)x^3 - (1/5)x^5 ] from 0 to 1
= 1/3 - 1/5
= 2/15
Similarly for x from -1 to 0.
Total area = 2/15 + 2/15 = 4/15
x^4 = x^2
x^4 - x^2 = 0
x²(x² - 1) = 0
x = -1, x = 0, x = 1
∫ (x^2 - x^4) dx
= [ (1/3)x^3 - (1/5)x^5 ] from 0 to 1
= 1/3 - 1/5
= 2/15
Similarly for x from -1 to 0.
Total area = 2/15 + 2/15 = 4/15
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intersection points between y = x^4 and y = x^2 are points -1, 0, 1 but both are even functions.
Area =
1
∫2(x^2 - x^4) dx =
0
1
| (2/3)x^3 - (2/5)x^5 = (2/3)*1^3 - (2/5)*1^5 - (2/3)*0^3 + (2/5)*0^5 = 2/3 - 2/5 = 4/15
0
Area =
1
∫2(x^2 - x^4) dx =
0
1
| (2/3)x^3 - (2/5)x^5 = (2/3)*1^3 - (2/5)*1^5 - (2/3)*0^3 + (2/5)*0^5 = 2/3 - 2/5 = 4/15
0
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just subtract the two antiderivatives