# Let R be the region bounded by the curves y = 1 - x^2 and y = x^6 - x + 1. Estimate the following quantities. (a) The x-coordinates of the points of the intersection of the curves. (b) The area...

Let R be the region bounded by the curves y = 1 - x^2 and y = x^6 - x + 1. Estimate the following quantities.

(a) The x-coordinates of the points of the intersection of the curves.

(b) The area of R

(c) The volume generated when R is rotated about the x-axis

(d) The volume generated when R is rotated about the y-axis

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Given curves are

y = 1 - x^2 ---------------------(1)

y = x^6 - x + 1 ---------------------(2)

**a)** To get the x-co ordinates of the points of intersection we equate 1 and 2

we obtain the expression as follows

**1** - x^2 = x^6 - x + **1**

=> - x^2 = x^6 - x

=> x^6 - x +x^2 =0

Tanking x common on LHS

x*(x^5 +x-1)=0

so the x-co ordinates are x=0 and x=0.754878 on simplification

**The x-coordinates and the intersection of the two curves can be seen in the attachments**

**b)** Area bounded

It is given as

`int_0^0.754878(x-x^2-x^6)dx`

`=[((x^2)/2)-((x^3)/3)-((x^7)/7)] `**0.754878** to **0**

= **0.121579** square units approx

**c)** volume generated when R is rotated about the x-axis

v=`int_0^0.754878 pi*((x-x^2 -x^6)^2)dx` = **0.07341** cubic units

**d)** volume generated when R is rotated about the y-axis

v=`int_0^0.754878 2*pi*x*((x-x^2-x^6))dx` = **0.30804** cubic units