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Let's solve each part of this question by finding the value of \( a \) that satisfies each integral equation.

### Problem Statement:
The task is to determine the value of \( a \) for each integral equation such that the given equality is satisfied.

### Solutions:

1. **Part (a):**
   \[
   \int_0^a x^{-\frac{1}{2}} \, dx = 4
   \]
   We solve by finding the antiderivative of \( x^{-\frac{1}{2}} \), setting it equal to 4, and solving for \( a \).

2. **Part (b):**
   \[
   \int_1^a x^{\frac{1}{2}} \, dx = 42
   \]
   Here, we find the antiderivative of \( x^{\frac{1}{2}} \), then evaluate from 1 to \( a \) and set the result equal to 42.

3. **Part (c):**
   \[
   \int_2^a (x - 2) \, dx = 4.5
   \]
   We find the antiderivative of \( x - 2 \), evaluate from 2 to \( a \), and set the result to 4.5.

4. **Part (d):**
   \[
   \int_2^a (x^3 - 16x) \, dx = -36
   \]
   This involves finding the antiderivative of \( x^3 - 16x \), evaluating from 2 to \( a \), and setting the result to -36.

Would you like detailed solutions for each part? 

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### Expanding Questions:
1. How can we determine the limits of integration for an unknown parameter in an integral?
2. What methods can be used to solve integrals involving powers and polynomial expressions?
3. How does changing the variable in the integrand affect the limits of integration?
4. In what scenarios does an integral result in a negative value, as seen in part (d)?
5. How can we verify the correctness of an integral solution?

### Tip:
For definite integrals, always evaluate the antiderivative at both limits carefully to avoid sign errors.

Tentukan nilai a sehingga tiap persamaan berikut ini dipenuhi.

Learn how to find the value of a that satisfies each given integral equation. This problem involves definite integrals, limits of integration, and applying the Fundamental Theorem of Calculus. Suitable for high school students in Grades 11-12.

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