The approximate value of \(\int_1^5 x^2 dx\) using trapezoidal rule with \(n = 4\) is

Question

The surface area of a spherical balloon is increasing at the rate of \( 2 \, \text{cm}^2/\text{sec} \). Then the rate of increase in the volume of the balloon, when the radius of the balloon is \( 6 \, \text{cm} \), is:

Answer 1

To approximate the value of the integral \(\int_1^5 x^2 dx\) using the trapezoidal rule with \(n = 4\), we first need to understand how the trapezoidal rule works.

The trapezoidal rule is used to approximate the value of a definite integral. The formula for the trapezoidal rule is:

\[ \int_a^b f(x) \, dx \approx \frac{h}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \]

where \(h\) is the width of each subinterval, and \(x_i\) are the endpoints of the subintervals.

Given:

Function \(f(x) = x^2\)
Lower limit \(a = 1\)
Upper limit \(b = 5\)
Number of subintervals \(n = 4\)

To calculate \(h\), the width of each subinterval, using:

\(h = \frac{b-a}{n} = \frac{5-1}{4} = 1\)

With \(h = 1\), the points \(x_0, x_1, x_2, x_3, x_4\) are:

\(x_0 = 1\)
\(x_1 = 2\)
\(x_2 = 3\)
\(x_3 = 4\)
\(x_4 = 5\)

Calculate \(f(x_i) = x_i^2\) for each point:

\(f(x_0) = 1^2 = 1\)
\(f(x_1) = 2^2 = 4\)
\(f(x_2) = 3^2 = 9\)
\(f(x_3) = 4^2 = 16\)
\(f(x_4) = 5^2 = 25\)

Substitute these values into the trapezoidal formula:

\[ \int_1^5 x^2 dx \approx \frac{1}{2} \left( 1 + 2(4 + 9 + 16) + 25 \right) \] \[ = \frac{1}{2} \left( 1 + 2 \times 29 + 25 \right) \] \[ = \frac{1}{2} \left( 1 + 58 + 25 \right) \] \[ = \frac{1}{2} \times 84 = 42 \]

Therefore, the approximate value of the integral \(\int_1^5 x^2 dx\) using the trapezoidal rule with \(n = 4\) is \(42\).

Thus, the correct answer is: 42.

Answer 2