Question

Write the summation to estimate the area under the curve y = 2x2 + 1 from x = 0 to x = 4 using 4 rectangles and left endpoints.

the summation from i equals 0 to 4 of the quantity 2 times i squared plus 1
the summation from i equals 1 to 3 of the quantity 2 times i squared plus 1
the summation from i equals 0 to 3 of the quantity 2 times i squared plus 1
the summation from i equals 1 to 4 of the quantity 2 times i squared plus 1

Answer 1

Hello,
Answer C:
the summation from i equals 0 to 3 of the quantity 2 times i squared plus 1

`\sum_(i=0)^3[(i+1-i)*(2*i^2+1)]\\\\ =\sum_(i=0)^3[(2*i^2+1)]\\\\`

Answer 2

The correct option is C.

Explanation:

The given equation of curve is

`y=2x^2+1`

We need to find the area under the given curve from x = 0 to x = 4 using 4 rectangles and left endpoints.

We have 4 rectangles in x = 0 to x = 4. So,

`\Delta x=1`

Left Riemann sum:

`\sum_(i=0)^(n-1)\Delta xf(x_i)`

Using Left Riemann sum, we get

`\sum_(i=0)^(4-1)(1)f(i)`

`\sum_(i=0)^(3)(2i^2+1)`
`[\because y=2x^2+1]`

The area under the given curve from x = 0 to x = 4 using 4 rectangles and left endpoints is

`\sum_(i=0)^(3)2i^2+1`

Therefore the correct option is C.