Evaluate Integral ∫5√y Dy [0 To 4] With Calculus Theorem

Hey guys! Today, we're diving into the fascinating world of calculus to tackle a classic problem: evaluating the definite integral ∫[0 to 4] 5√y dy. We'll be using the Fundamental Theorem of Calculus, Part 1, which is a cornerstone concept in calculus. So, grab your thinking caps, and let's get started!

Understanding the Fundamental Theorem of Calculus, Part 1

Before we jump into the problem, let's quickly recap what the Fundamental Theorem of Calculus, Part 1, is all about. In essence, it provides a powerful method for evaluating definite integrals. It tells us that if we have a continuous function f(x) on an interval [a, b], and F(x) is any antiderivative of f(x) (meaning F'(x) = f(x)), then:

∫[a to b] f(x) dx = F(b) - F(a)

In simpler terms, to evaluate a definite integral, we need to find the antiderivative of the function, plug in the upper and lower limits of integration, and subtract the results. Easy peasy, right? This theorem bridges the gap between differentiation and integration, showing they are inverse processes. This is super useful because finding the area under a curve, a typical integration problem, can be solved by finding an antiderivative, which is often easier than directly summing up infinitely thin rectangles. For students and professionals alike, mastering this theorem is crucial for solving a wide range of problems in physics, engineering, and economics, where integrals are frequently used to model various phenomena.

Breaking Down the Integral ∫[0 to 4] 5√y dy

Okay, now let's apply this to our specific problem. We have the integral ∫[0 to 4] 5√y dy. Here, our function f(y) is 5√y, and our limits of integration are 0 and 4. Our mission is to find the antiderivative of 5√y and then use the Fundamental Theorem to evaluate the integral. This process involves a few key steps: first, rewriting the integrand to make it easier to work with; second, applying the power rule for integration; and third, evaluating the antiderivative at the limits of integration. Each step is crucial for arriving at the correct answer, so let's take our time and do it right.

Step-by-Step Solution

Let’s break down the solution step-by-step so you can follow along and understand each part of the process.

Step 1: Rewrite the Integrand

The first thing we want to do is rewrite 5√y in a more convenient form for integration. Remember that √y is the same as y^(1/2). So, we can rewrite our function as 5y^(1/2). This makes it much easier to apply the power rule for integration. Rewriting expressions is a common trick in calculus, allowing us to transform complex-looking functions into simpler, more manageable forms. By expressing the square root as a fractional exponent, we set ourselves up for the next step, which involves applying a basic integration rule.

Step 2: Find the Antiderivative

Now comes the fun part – finding the antiderivative! We'll use the power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. Applying this rule to 5y^(1/2), we get:

∫5y^(1/2) dy = 5 * ∫y^(1/2) dy = 5 * [y^(1/2 + 1) / (1/2 + 1)] + C

Simplifying this, we have:

5 * [y^(3/2) / (3/2)] + C = (10/3) * y^(3/2) + C

So, our antiderivative F(y) is (10/3)y^(3/2) + C. Remember, the constant of integration, C, is crucial for indefinite integrals. However, when evaluating definite integrals, the C terms will cancel out, so we often omit them for simplicity. Understanding and applying the power rule is fundamental to integral calculus, and it's a skill that will serve you well in many different contexts.

Step 3: Apply the Fundamental Theorem

Alright, we're in the home stretch! Now we apply the Fundamental Theorem of Calculus, Part 1. We need to evaluate our antiderivative F(y) = (10/3)y^(3/2) at the upper and lower limits of integration, which are 4 and 0, respectively. Then, we subtract the value at the lower limit from the value at the upper limit:

∫[0 to 4] 5√y dy = F(4) - F(0) = [(10/3) * 4^(3/2)] - [(10/3) * 0^(3/2)]

Let's break this down further. 4^(3/2) means we first take the square root of 4, which is 2, and then raise it to the power of 3: 2^3 = 8. And, of course, 0^(3/2) is just 0. So we have:

[(10/3) * 8] - [(10/3) * 0] = (80/3) - 0 = 80/3

Therefore, the value of the integral ∫[0 to 4] 5√y dy is 80/3. This step demonstrates the practical application of the Fundamental Theorem, turning the abstract concept of integration into a concrete numerical result.

Final Answer

So, there you have it! We've successfully evaluated the integral ∫[0 to 4] 5√y dy using the Fundamental Theorem of Calculus, Part 1. The final answer is 80/3. Remember, the key steps were rewriting the integrand, finding the antiderivative using the power rule, and then applying the Fundamental Theorem to evaluate the definite integral. This process not only gives us the numerical value of the integral but also deepens our understanding of the relationship between differentiation and integration.

Why This Matters

You might be wondering,