Question

Question attached as screenshot belowPlease help me with my Calc hw

Answer 1

Step 1: State what is given in the question

Given the area under a function with the upper and lower limit estimated using four approximating rectangles and left endpoint.

Step 2: State what is to be determined

We are to determine if the estimation would be underestimation or overestimation

In other to achieve this, we would first define left end point approximation.

In other to find the area under a curve using partition. This can either be left endpoint approximation or right endpoint approximation

Our focus is left endpoint approximation where on each subinterval of the partion

`\begin{gathered} (x_(i-1),x_i) \\ i=1,2,3,4,\ldots \end{gathered}`

We would construct a rectangle with width Δx and height equal to

`f(x_(i-1))`

which is the function value at the left endpoint of the subinterval.

Then the area of this rectangle is

`f(x_(i-1))\Delta x`

Adding the areas of all these rectangles, we get an approximate value for the area under under the curve. The diagram below shows how the shape would be

Taking a careful look at the rectangles under the curve, the areas of the rectangles would give a less value than the area under the entire curve. This is because there are some spaces between the curve and the rectangles

Hence, the area under the curve using four approximating rectangles and left endpoints would be an underestimate of the area under the curve