The continuous function $f$ is graphed.

We're interested in the area under the curve between $x=-5$ and $x=3$, and we're considering using Riemann sums to approximate it.

Order the areas from least (on top) to greatest (on bottom).

Actual area under the curve
Right Riemann sum
Left Riemann sum

Question

The continuous function $f$ is graphed.

We're interested in the area under the curve between $x=-5$ and $x=3$, and we're considering using Riemann sums to approximate it.

Order the areas from least (on top) to greatest (on bottom).

Actual area under the curve
Right Riemann sum
Left Riemann sum

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To approximate the area under the curve between $x = -5$ and $x = 3$ using Riemann sums, we first need to understand the definitions of the left Riemann sum, right Riemann sum, and the actual area under the curve. ‖ ‖ step 2 ⋮ The left Riemann sum approximates the area under the curve by summing the areas of rectangles whose heights are determined by the function values at the left endpoints of subintervals. ‖ ‖ step 3 ⋮ The right Riemann sum approximates the area under the curve by summing the areas of rectangles whose heights are determined by the function values at the right endpoints of subintervals. ‖ ‖ step 4 ⋮ The actual area under the curve is the integral of the function $f(x)$ from $x = -5$ to $x = 3$. This can be represented as $\int_{-5}^{3} f(x) \, dx$. ‖ ‖ step 5 ⋮ Typically, for a continuous and increasing function, the left Riemann sum will underestimate the area, and the right Riemann sum will overestimate the area. Therefore, the order from least to greatest is: ‖ ‖ step 6 ⋮ 1. Left Riemann sum ‖ ‖ step 7 ⋮ 2. Actual area under the curve ‖ ‖ step 8 ⋮ 3. Right Riemann sum ‖ ∻Answer∻ ⚹ Left Riemann sum, Actual area under the curve, Right Riemann sum ⚹ ∻Key Concept∻ ⚹Riemann Sums⚹ ∻Explanation∻ ⚹Riemann sums are methods for approximating the total area under a curve on a graph, otherwise known as the integral of a function. The left Riemann sum uses the left endpoints of subintervals, while the right Riemann sum uses the right endpoints.⚹◊What is the formula for calculating the area under a curve using Riemann sums?⍭ Generate me a similar question◊

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