Question

The graph of f(x) = 8 - x and line l which is tangent to f at x =1 is
shown in the figure to the right. The equation for line l is y =-3x +10.
Region R is the shaded region between line l, the graph off and the
y-axis while S is the shaded region between line l, the graph off and
the x-axis.
(a) Find the area of the shaded region R.

Answer 1

By taking the integral of the difference between the two given lines, we will see that R = 0.75.

How to find the area of the region R?

Notice that the area will be given by the integral of the difference between the line L and our function on the interval [0, 1]

Then we just need to compute:

$R = \int\limits^1_0 {-3x + 10 - (8 - x^3)} \, dx$

Solving that we will get:

$R = \int\limits^1_0 {-3x + 10 - (8 - x^3)} \, dx = \int\limits^1_0 ({-3x + 10 - 8 + x^3} )dx\\\\\R = \int\limits^1_0 ({-3x + 2 + x^3} )dx$

$R = \int\limits^1_0 ({-3x + 2 + x^3} )dx\\\\R = \frac{-3}{2}*(1)^2 + 2*1 + \frac{1^4}{4} = 0.75$

So the area of region R is 0.75 square units.

If you want to learn more about integrals, you can read:

https://brainly.com/question/14502499