Question

A curve is passing through the points $\left( {1,2} \right),\left( {1.5,2.4} \right),\left( {2,2.7} \right),\left( {2.5,2.8} \right),\left( {3,3} \right)$ then the area bounded by the curve, $X$-axis and $x = 1,x = 3$ using Simpson’s rule is
A) 5
B) $5.1$
C) $5.2$
D) $5.4$

Answer 1

We will first consider the given points. We need to find the area bounded by the curve. We will first write all the points in the form $f\left( x \right) = y$ and name them as ${y_1},{y_2},..$ and so on. Then we will find the value of $h$ which is $0.5$ and $n = 5$ that is the number of parts the interval is divided into. Next, we will apply the Simpson’s formula that is $A = \dfrac{h}{3}\left[ {{y_0} + 2\left( {{y_2}} \right) + 4\left( {{y_1} + {y_3}} \right) + {y_4}} \right]$. After substituting the values, we will simplify the equation to determine the area bounded by the curve.

Complete step by step solution:
We will first let the curve as $y = f\left( x \right)$.
The objective is to find the area bounded by the curve.
Now, we will write all the points in the form $f\left( x \right) = y$.
Thus, we get,

$$f\left( 1 \right) = 2 = {y_0} \\\ f\left( {1.5} \right) = 2.4 = {y_1} \\\ f\left( 2 \right) = 2.7 = {y_2} \\\ f\left( {2.5} \right) = 2.8 = {y_3} \\\ f\left( 3 \right) = 3 = {y_4} \\\$$

Next, we will find the value of $h$ using $h = b - a$.
Here, we have $h$ as $0.5$
Thus, we will use the formula of Simpson’s rule to find the area that is $A = \dfrac{h}{3}\left[ {{y_0} + 2\left( {{y_2}} \right) + 4\left( {{y_1} + {y_3}} \right) + {y_4}} \right]$.
Hence, we will substitute the values in the above formula and we get,

$$\Rightarrow A = \dfrac{{0.5}}{3}\left[ {2 + 2\left( {2.7} \right) + 4\left( {2.4 + 2.8} \right) + 3} \right] \\\ \Rightarrow A = \dfrac{{0.5\left( {31.2} \right)}}{3} \\\ \Rightarrow A = 5.2 \\\$$

Hence, we can conclude that the area bounded by the curve is $5.2$.

Thus, option C is correct.

Note:
Remember the formula of area of the curve using Simpson’s rule for the given number of grid points. Write all the coordinates carefully in the form $f\left( x \right) = y$ and name them accordingly. Value of $h$ can be obtained by substituting the value in the formula of $h$. While simplifying the expression, first open the brackets and then do the further calculations.