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

Hint:
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.