Question

For a fixed \(c \in \mathbb{R}\), let \(\alpha = \int_0^c (9x^2 - 5cx^4)dx\).
If the value of \(\int_0^c (9x^2 - 5cx^4)dx\) obtained by using the Trapezoidal rule is equal to \(\alpha\), then the value of c is .................. (rounded off to 2 decimal places).

Hint:

The error of the simple Trapezoidal rule is zero if and only if the integral of the second derivative of the function over the interval is zero in a specific weighted sense. A simpler condition is when the exact value equals the approximation, which leads to an algebraic equation. Always start with this direct algebraic approach.

Answer 1

Note: The provided answer key for this question is a range from 0.24 to 0.26. However, a direct mathematical interpretation of the problem yields a different result. The solution below follows the direct interpretation. The discrepancy suggests a potential error in the problem statement or the provided answer key.
Step 1: Understanding the Concept:
The problem states that the exact value of a definite integral is equal to the value obtained using the (simple) Trapezoidal rule. We need to find the value of the upper limit of integration, \(c\), for which this is true.
Step 2: Key Formula or Approach:
1. Calculate the exact value of the integral, \(\alpha\).
2. Calculate the approximate value using the single-interval Trapezoidal rule formula: \(\int_a^b f(x)dx \approx \frac{b-a}{2}[f(a) + f(b)]\).
3. Set the two expressions equal to each other and solve for \(c\).
Step 3: Detailed Calculation:
Let \(f(x) = 9x^2 - 5cx^4\).
Step 3.1: Calculate the exact value \(\alpha\). \[ \alpha = \int_0^c (9x^2 - 5cx^4)dx = \left[ \frac{9x^3}{3} - \frac{5cx^5}{5} \right]_0^c = [3x^3 - cx^5]_0^c \] \[ \alpha = (3c^3 - c(c^5)) - 0 = 3c^3 - c^6 \] Step 3.2: Calculate the Trapezoidal rule approximation.
The interval is \([a, b] = [0, c]\). \[ T = \frac{c-0}{2}[f(0) + f(c)] \] First, evaluate the function at the endpoints: \[ f(0) = 9(0)^2 - 5c(0)^4 = 0 \] \[ f(c) = 9c^2 - 5c(c^4) = 9c^2 - 5c^5 \] Now, substitute these into the Trapezoidal rule formula: \[ T = \frac{c}{2}[0 + (9c^2 - 5c^5)] = \frac{9c^3 - 5c^6}{2} \] Step 3.3: Set \(T = \alpha\) and solve for c. \[ \frac{9c^3 - 5c^6}{2} = 3c^3 - c^6 \] Multiply both sides by 2: \[ 9c^3 - 5c^6 = 6c^3 - 2c^6 \] Rearrange the terms to one side: \[ (9c^3 - 6c^3) = (5c^6 - 2c^6) \] \[ 3c^3 = 3c^6 \] Divide by 3: \[ c^3 = c^6 \implies c^6 - c^3 = 0 \] Factor out \(c^3\): \[ c^3(c^3 - 1) = 0 \] This gives two possible solutions: \(c^3 = 0 \implies c=0\) or \(c^3 - 1 = 0 \implies c^3 = 1 \implies c = 1\). Given that \(c\) is a "fixed" value for which an integral is defined, the non-trivial solution \(c=1\) is the intended answer.
Step 4: Final Answer:
The value of c is 1. Rounded to 2 decimal places, this is 1.00.
Step 5: Why This is Correct:
The solution equates the analytical expression for the integral with the expression from the Trapezoidal rule, as per the problem statement. Solving the resulting algebraic equation for \(c\) yields the unique non-zero solution \(c=1\).