Calculate Integreal 8-X Dx in Two Ways

Calculating the integral of 8-x dx involves finding the area under the curve of the function f(x) = 8 - x between two points. This can be done using two primary methods: direct integration and the substitution method. Both approaches will yield the same result but may differ in complexity and ease of application.

Method 1: Direct Integration

The direct integration method involves integrating the function term by term. For the integral of 8 - x dx, we can separate it into two simpler integrals:

∫(8 - x) dx = ∫8 dx - ∫x dx

To solve these integrals, we use the basic integration rules:

The integral of a constant is the constant multiplied by x.
The integral of x is (x²)/2.

Applying these rules:

∫8 dx = 8x + C₁

∫x dx = (x²)/2 + C₂

Combining these results and adding the constants of integration (which cancel out when we evaluate the definite integral):

∫(8 - x) dx = 8x - (x²)/2 + C

For a definite integral from a to b, we evaluate the antiderivative at the upper and lower limits:

∫[a to b] (8 - x) dx = [8b - (b²)/2] - [8a - (a²)/2]

This gives us the area under the curve between points a and b.

Method 2: Substitution Method

The substitution method is useful when the integrand is a composite function. For the integral of 8 - x dx, we can use substitution to simplify the calculation.

Let's set u = 8 - x. Then, du = -dx, which implies dx = -du. This changes the integral:

∫(8 - x) dx = ∫u (-du) = -∫u du

Now, we can integrate u:

-∫u du = - (u²)/2 + C

Substituting back u = 8 - x:

- (u²)/2 + C = - (8 - x)²/2 + C

Expanding this expression gives us the same result as the direct integration method:

- (8 - x)²/2 + C = - (64 - 16x + x²)/2 + C = -32 + 8x - (x²)/2 + C

For a definite integral from a to b, we evaluate the antiderivative at the upper and lower limits:

∫[a to b] (8 - x) dx = [- (8 - b)²/2] - [- (8 - a)²/2]

This approach is particularly useful when dealing with more complex functions where substitution simplifies the integrand.

Comparison of Methods

Both methods yield the same result for the integral of 8 - x dx. The direct integration method is straightforward for this simple function, while the substitution method demonstrates a technique that can be applied to more complex integrals.

For simple polynomials like 8 - x, direct integration is often the most efficient method. However, substitution becomes valuable when dealing with composite functions or integrals that require a change of variables.

Understanding both methods provides a comprehensive approach to solving integrals, allowing you to choose the most appropriate technique based on the function's complexity.

FAQ

What is the integral of 8 - x dx?: The integral of 8 - x dx is 8x - (x²)/2 + C, where C is the constant of integration. This represents the antiderivative of the function 8 - x.
How do I evaluate a definite integral of 8 - x dx from a to b?: To evaluate the definite integral, substitute the upper limit (b) and lower limit (a) into the antiderivative and subtract the two results: [8b - (b²)/2] - [8a - (a²)/2].
When would I use the substitution method instead of direct integration?: The substitution method is particularly useful when the integrand is a composite function or when a substitution simplifies the integral, such as in integrals involving trigonometric or exponential functions.
Can I use the integral of 8 - x dx to find the area under the curve?: Yes, the integral of 8 - x dx gives the area under the curve of the function f(x) = 8 - x between the specified limits. The result represents the net area, which can be positive or negative depending on the function's behavior.