`y = root(3)(x), 0<=x<=27` Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

Textbook Question

Chapter 5, 5.3 - Problem 49 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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Refer the graph in the attached image.

From the graph it appears that area of the region is `~~` 70% of the rectangle.

Area of the region=`~~` 70/100(Area of Rectangle)

Area of the region =`~~0.7(27*3)~~56.7`

Exact Area of the region=`int_0^27root(3)(x)dx`

`=int_0^27x^(1/3)dx`

`=[x^(1/3+1)/(1/3+1)]_0^27`

`=[3/4x^(4/3)]_0^27`

`=[3/4(27)^(4/3)]-[3/4(0)^(4/3)]`

`=(3/4(3^3)^(4/3))`

`=(3/4(3^4))`

`=3/4(81)`

=60.75

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