Q. 3 - Calculus

Question

In the reading we used objects whose area we knew (squares) to approximate the area of a more complicated object (a circle). The same kind of technique can be used to approximate the area of the blob pictured here 

(a) Given that each of the squares in the grid has a side length of  $\frac{1}{2}$unit, approximate the area of the blob.

 (b) How could you get a better approximation?

Step-by-Step Solution

Answer

a) The approximated area is $15.5$ square units.

b) By using a smaller squares grid we can get a better approximation of the area.

Step 1. Given information

Given diagram:

Side length of each square is $\frac{1}{2}$ units

.

Part (a) step 1. Find approximated area of the shaded region.

In the given figure there are 46 complete squares and 32 partial squares.

So the approximated area of the shaded region is:

$$\begin{gathered} A={\left(\frac{1}{2}\right)}^2\times 46+\frac{1}{2}\times {\left(\frac{1}{2}\right)}^2\times 32 \\ =\frac{23}{2}+4 \\ =\frac{31}{2} \\ =15.5 \end{gathered}$$

So the approximated area of the shaded region is 15.5 square units.

Part (b) Step 1. Explanation of getting an approximation

We can get a better approximation of the area by using small squares instead of large squares.