Estimate ∫₁⁴ √(4𝓍 + 1) d𝓍 by evaluating the left, right, and midpoint Riemann sums using a regular partition with n = 6 subintervals.

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Divide the interval [1, 4] into n = 6 subintervals. Calculate the width of each subinterval, Δ𝓍, using the formula Δ𝓍 = (b - a) / n, where a = 1 and b = 4.

Determine the endpoints of each subinterval. These will be the points 𝓍₀, 𝓍₁, ..., 𝓍₆, where 𝓍₀ = 1 and 𝓍₆ = 4, and the intermediate points are spaced by Δ𝓍.

For the left Riemann sum, evaluate the function √(4𝓍 + 1) at the left endpoints of each subinterval (𝓍₀, 𝓍₁, ..., 𝓍₅). Multiply each function value by Δ𝓍 and sum them together.

For the right Riemann sum, evaluate the function √(4𝓍 + 1) at the right endpoints of each subinterval (𝓍₁, 𝓍₂, ..., 𝓍₆). Multiply each function value by Δ𝓍 and sum them together.

For the midpoint Riemann sum, calculate the midpoint of each subinterval, which is given by (𝓍ᵢ + 𝓍ᵢ₊₁) / 2 for i = 0, 1, ..., 5. Evaluate the function √(4𝓍 + 1) at each midpoint, multiply each function value by Δ𝓍, and sum them together.

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Riemann Sums

Riemann sums are a method for approximating the definite integral of a function over an interval. They involve partitioning the interval into subintervals and summing the areas of rectangles formed by evaluating the function at specific points within each subinterval. The left, right, and midpoint Riemann sums use the left endpoint, right endpoint, and midpoint of each subinterval, respectively, to determine the height of the rectangles.

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as ∫_a^b f(x) dx, where 'a' and 'b' are the limits of integration. The definite integral can be interpreted as the limit of Riemann sums as the number of subintervals approaches infinity, providing a precise value for the area under the curve.

Regular Partition

A regular partition divides an interval into equal subintervals, which simplifies the calculation of Riemann sums. In this case, with n = 6 subintervals over the interval [1, 4], each subinterval will have a width of (4-1)/6 = 0.5. This uniformity allows for straightforward computation of the function values at the specified points (left, right, or midpoint) for each subinterval.