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(a) Estimate the area under the graph of f(x)=5sqrtx from x=0 to x=4 using four approximating rectangles and right endpo R_(4)="-1" Enter a number. Sketch the graph and the rectangles.

(a) Estimate the area under the graph of f(x)=5x f(x)=5 \sqrt{x} from x=0 x=0 to x=4 x=4 using four approximating rectangles and right endpo\newlineR4undefined−1 R_{4} \xlongequal{-1} \newlineEnter a number.\newlineSketch the graph and the rectangles.

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Q. (a) Estimate the area under the graph of f(x)=5x f(x)=5 \sqrt{x} from x=0 x=0 to x=4 x=4 using four approximating rectangles and right endpo\newlineR4undefined−1 R_{4} \xlongequal{-1} \newlineEnter a number.\newlineSketch the graph and the rectangles.
  1. Calculate Width of Rectangles: Divide the interval [0,4][0, 4] into 44 equal subintervals to find the width of each rectangle.\newlineWidth of each rectangle (Δx)(\Delta x) = (4−0)/4=1(4 - 0) / 4 = 1.
  2. Determine Right Endpoints: Determine the right endpoints of each subinterval: x1=1x_1 = 1, x2=2x_2 = 2, x3=3x_3 = 3, x4=4x_4 = 4.
  3. Calculate Heights of Rectangles: Calculate the height of each rectangle using the right endpoints and the function f(x)=5xf(x) = 5\sqrt{x}.\newlineHeight at x1x_1: f(1)=51=5f(1) = 5\sqrt{1} = 5.\newlineHeight at x2x_2: f(2)=52f(2) = 5\sqrt{2}.\newlineHeight at x3x_3: f(3)=53f(3) = 5\sqrt{3}.\newlineHeight at x4x_4: f(4)=54=5×2=10f(4) = 5\sqrt{4} = 5 \times 2 = 10.
  4. Compute Areas of Rectangles: Compute the area of each rectangle (Area = width ×\times height).\newlineArea of rectangle 11: A1=Δx×f(1)=1×5=5A_1 = \Delta x \times f(1) = 1 \times 5 = 5.\newlineArea of rectangle 22: A2=Δx×f(2)=1×52A_2 = \Delta x \times f(2) = 1 \times 5\sqrt{2}.\newlineArea of rectangle 33: A3=Δx×f(3)=1×53A_3 = \Delta x \times f(3) = 1 \times 5\sqrt{3}.\newlineArea of rectangle 44: A4=Δx×f(4)=1×10=10A_4 = \Delta x \times f(4) = 1 \times 10 = 10.
  5. Estimate Total Area: Add up the areas of the rectangles to estimate the total area under the curve.\newlineTotal area ≈A1+A2+A3+A4=5+52+53+10\approx A_1 + A_2 + A_3 + A_4 = 5 + 5\sqrt{2} + 5\sqrt{3} + 10.
  6. Simplify Total Area Expression: Simplify the expression for the total area.\newlineTotal area ≈5+5(2+3)+10\approx 5 + 5(\sqrt{2} + \sqrt{3}) + 10.
  7. Calculate Numerical Total Area: Calculate the numerical value of the total area.\newlineTotal area ≈5+5(1.414+1.732)+10\approx 5 + 5(1.414 + 1.732) + 10.\newlineTotal area ≈5+5(3.146)+10\approx 5 + 5(3.146) + 10.\newlineTotal area ≈5+15.73+10\approx 5 + 15.73 + 10.\newlineTotal area ≈30.73\approx 30.73.

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