Question

If the trapezoidal rule with a single interval [0, 1] is exact for the approximate value of \( \int_0^1 (x^3 - cx^2) \, dx \). Then the value of \( c \).

• A. \( 2/3 \)
• B. \( 3/2 \)
• C. \( 1 \)
• D. \( 0 \)

Hint:

The trapezoidal rule is exact for polynomials of degree at most 1. For it to be exact for a higher degree polynomial, the error term of the trapezoidal rule must be zero. The error term is related to the second derivative of the function. However, the direct approach of equating the trapezoidal approximation to the exact integral is straightforward here.

Answer 1

The Correct Option is B

Step 1: Apply the trapezoidal rule with a single interval [0, 1].
For a single interval \([a, b]\), the trapezoidal rule for approximating the integral \( \int_a^b f(x) \, dx \) is given by: \[ \int_a^b f(x) \, dx \approx \frac{b - a}{2} [f(a) + f(b)] \] In this case, \( a = 0 \), \( b = 1 \), and \( f(x) = x^3 - cx^2 \). So, \[ \int_0^1 (x^3 - cx^2) \, dx \approx \frac{1 - 0}{2} [f(0) + f(1)] \] \[ \int_0^1 (x^3 - cx^2) \, dx \approx \frac{1}{2} [(0^3 - c(0)^2) + (1^3 - c(1)^2)] \] \[ \int_0^1 (x^3 - cx^2) \, dx \approx \frac{1}{2} [0 + (1 - c)] \] \[ \int_0^1 (x^3 - cx^2) \, dx \approx \frac{1 - c}{2} \] Step 2: Calculate the exact value of the integral.
\[ \int_0^1 (x^3 - cx^2) \, dx = \left[ \frac{x^4}{4} - \frac{cx^3}{3} \right]_0^1 \] \[ \int_0^1 (x^3 - cx^2) \, dx = \left( \frac{1^4}{4} - \frac{c(1)^3}{3} \right) - \left( \frac{0^4}{4} - \frac{c(0)^3}{3} \right) \] \[ \int_0^1 (x^3 - cx^2) \, dx = \frac{1}{4} - \frac{c}{3} - 0 \] \[ \int_0^1 (x^3 - cx^2) \, dx = \frac{1}{4} - \frac{c}{3} \] Step 3: Set the approximate value equal to the exact value since the trapezoidal rule is exact.
\[ \frac{1 - c}{2} = \frac{1}{4} - \frac{c}{3} \] Step 4: Solve for \( c \).
Multiply both sides by 12 to eliminate the fractions: \[ 12 \times \frac{1 - c}{2} = 12 \times \left( \frac{1}{4} - \frac{c}{3} \right) \] \[ 6(1 - c) = 3(1) - 4(c) \] \[ 6 - 6c = 3 - 4c \] Rearrange the terms to solve for \( c \): \[ 6 - 3 = 6c - 4c \] \[ 3 = 2c \] \[ c = \frac{3}{2} \] Step 5: Select the correct answer.
The value of \( c \) is \( 3/2 \), which corresponds to option 2.