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Question
Find the area under the given curve and given line:
y = x4, x = 1, x = 5 and x-axis
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Solution
The curve y = x4 passes through the point (0, 0). The line OY is symmetric.
Now, y = x4
`dy/dx = 4x^3`
The sign of `dy/dx` changes from -ve to +ve when x moves through x = 0.
∴ x = 0 is the lowest point.
∴ Area of the region bounded by y = x4, x = 1, x = 5 and x-axis
= Area of the region PABQ
`= int_1^5 y dx = int_1^5 x^4 dx`
`= [x^5/5]_1^5 = [5^5/5 - 1/5]`
`= [5^4 - 1/5]`
`= 625 - 1/5`
`= (3125 - 1)/5`
`= 3124/5`
= 624.8 square unit
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