Question

Use the left-hand rule to estimate the area under the graph of y = (x + 1)2 – 1 for values of x from 0 to 3.

Answer 1

11 square units

Explanation:

Using the left-hand rule to estimate the area under a graph y = f(x) between the interval x = a to x = b is given by

`A_(n) =(b-a)/(n)[f(0) + f(1) + f(2) + ........ + f(n-1)]` .......... (1)

{Where n is the number of rectangles we want to make the area under curve.}

Now, in out case a = 0 and b = 3 and
`f(x) = (x - 1)^(2)  -1` ....... (2)

Let us assume that n = 3, then there will be 3 rectangles.

Therefore, from the equation (1), we can write

`A_(3) =(3-0)/(3)[f(0) + f(1) + f(2)]`

⇒
`A_(n) =(3-0)/(3)[0 + 3 + 8]` = 11 square units

{Since we have from equation (2), f(0) = 0, f(1) = 3 and f(2) = 8} (Answer)