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The Fundamental Theorem of Calculus is a powerful tool that links the concept of differentiation with integration. It consists of two main parts.

• Part 1 tells us that if a function is continuous over an interval, then the function has an antiderivative, which means we can find the area beneath its graph more easily.
• Part 2, however, is particularly relevant in solving problems like our original exercise. It helps compute the exact area under a curve using definite integrals.

In our exercise, using Part 2, we determine the area under the curve of the function \( y = x^2 \) from \( x = 0 \) to \( x = 4 \). By computing the integral \( \int_{0}^{4} x^2 \, dx \), the Fundamental Theorem of Calculus aids in translating the function into a solved expression: \( \left[ \frac{x^3}{3} \right]_{0}^{4} = \frac{64}{3} \). This result gives us the exact computation of the area.

The Left-endpoint Riemann sum is a method for estimating the area under a curve by using rectangles to approximate the space beneath a function's graph. For our task, we divide our interval

• Here, each rectangle uses its leftmost point on the curve as the height.
• We divide the interval into parts. With \(N = 10\) rectangles, and each rectangle has a width of \( \Delta x = 0.4 \).

We calculate the total area by summing up the areas of these rectangles: \[ L_{10} = \sum_{i=0}^{9} f(x_i) \cdot 0.4 \] where \(x_i\) starts from 0 and increases by the rectangle width of 0.4 each time. Calculating each rectangle's area involves substituting the x-values into \( f(x) = x^2 \). The culmination of these calculations offers a tangible yet approximate figure of the targeted area.

The Right-endpoint Riemann sum determines the approximate area under a curve with rectangles as well, but is distinct in its approach. Instead of using the leftmost point, the rightmost point on the curve is used as the rectangle's height.

• Just like the left-endpoint sum, the interval is divided into parts and we have \(N = 10\) rectangles with each having a width of \( \Delta x = 0.4 \).
• This method computes using x-points that start from 0.4 to 4 because each rectangle's right edge is the sample height point.

For our function \( y = x^2 \), we sum: \[ R_{10} = \sum_{i=1}^{10} f(x_i) \cdot 0.4 \] These sums offer a valuable approximation, yet it's noteworthy that either Riemann sum gives a unique perspective on the area's estimate.

Integral calculus is at the heart of calculating areas beneath curves. This branch of calculus revolves around integrals, which are fundamental in determining quantities like areas, volumes, and even probabilities for complex functions.

• Indefinite integrals are like reverse derivatives, providing an antiderivative or a "family of functions".
• Definite integrals, however, result in actual numbers, giving precise answers for areas under curves between specific points, as in our exercise.

The exact area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 4 \), was found through the definite integral: \[ \int_{0}^{4} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{4} = \frac{64}{3} \]Integral calculus and its methods provide not just approximation but sheer calculation precision, shifting from estimated values to exact figures. Through the step-by-step computation, both Riemann sums and definite integrals bolster your understanding of how calculus interprets such mathematical phenomena.