91Ó°ÊÓ

Finding the area under a curve involves calculating a definite integral. A definite integral, unlike an indefinite one, provides a specific numerical value. This value represents the area between the curve and the x-axis within the specified limits. In this exercise, we're working with the curve represented by the function \(y = 5\sqrt{9-x^2}\), and we need to find the area from \(x = -3\) to \(x = 3\).
The setup for this problem is an integral: \(\int_{-3}^{3} 5\sqrt{9-x^2} \, dx\), which helps us calculate the exact area under the curve between these two points. The limits of the integral \([-3, 3]\) define the region on the x-axis we're interested in.
The main goal with definite integrals is to receive a finite value that tells us the total accumulated area, which is what we interpret as the area "under the curve."

• The definite integral includes limits of integration.
• It results in a numerical value fixing the area.
• It is essential for problems involving physical quantities like area, volume, and more.

Integration by substitution is a method that simplifies complex integrals by transforming them into a simpler form, often resembling simpler integrals that are easier to solve. For this exercise, substitution helps manage the integral's radical expression.
We need to integrate \(5\sqrt{9-x^2}\), and a suitable substitution can simplify the process. By setting \(x = 3\sin(u)\), and computing \(dx = 3\cos(u) \, du\), the integral is transformed:

• Change of variable: \(x = 3\sin(u)\)
• Differential transformation: \(dx = 3\cos(u) \, du\)
• New integral form: \(\int_{-rac{\pi}{2}}^{\frac{\pi}{2}} 5\sqrt{9-(3\sin(u))^2} \cdot 3\cos(u) \, du\)

Substitution aids in managing the limits by translating them from \(x\) to \(u\), ensuring our integral remains accurate and much easier to compute.

Trigonometric substitution is a technique used specifically to deal with integrals containing square roots and quadratic expressions. It involves substituting variables of trigonometric functions to simplify these expressions.
For the integral \(\int_{-3}^{3} 5\sqrt{9-x^2} \, dx\), using \(x = 3\sin(u)\) leverages the identity \(1-\sin^2(u) = \cos^2(u)\). This turns tricky expressions into straightforward trigonometric identities.

• Helps simplify complex expressions under the square root.
• Utilizes trigonometric identities to reduce complexity.
• Changes the range of integration to reflect the substitution.

By translating \(9-x^2\) into \(9\cos^2(u)\), the integration process becomes less cumbersome, transforming into a simpler integral over trigonometric functions.

The cosine reduction formula is a strategy used to compute the integral of powers of the cosine function. It comes in handy when dealing with integrals like \(\int \cos^n(u) \, du\).
In this problem, we face \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 45\cos^3(u) \, du\), which is an ideal candidate for the cosine reduction formula. The formula for \(n = 3\) gives us:
\[\int \cos^3(u) \, du = \frac{1}{3} \cos^2(u) \sin(u) + \frac{2}{3} \int \cos(u) \, du\]

• Applies when integrating powers of cosine.
• Breaks down the problem by reducing the power iteratively.
• Transforms integrals into more manageable parts.

This approach simplifies evaluating \(\int \cos^3(u) \, du\), breaking it into parts that are straightforward to integrate and evaluate at the boundaries, resulting in the desired area calculation.