Sketch and clearly indicate the area enclosed by the graph y = -2x² + 2 the lines x = -1 and x = 1. Also clearly show the representative strip that will be used to calculate the area enclosed.

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Asked on July 6, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-07-06T06:10:44.898Z

here are the solutions for the area calculation. QUESTION 4.3 4.3.1 Sketch and clearly indicate the area enclosed by the graph y = -2x^2 + 2, the lines x = -1 and x = 1. Also clearly show the representative strip that will be used to calculate the area enclosed. Step 1: Identify the type of graph and key points. The equation y = -2x^2 + 2 represents a parabola opening downwards. • The y-intercept is found by setting x=0: y = -2(0)^2 + 2 = 2. So, the point is (0, 2). This is also the vertex. • The x-intercepts are found by setting y=0: 0 = -2x^2 + 2 2x^2 = 2 x^2 = 1 x = ± 1. So, the points are (-1, 0) and (1, 0). Step 2: Describe the sketch. Draw a coordinate plane. Plot the vertex at (0, 2) and the x-intercepts at (-1, 0) and (1, 0). Draw a smooth parabola opening downwards through these points. The lines x = -1 and x = 1 are vertical lines that pass through the x-intercepts. The area enclosed is the region under the parabola and above the x-axis, between x=-1 and x=1. To show a representative strip, draw a thin vertical rectangle within this enclosed area, from the x-axis up to the curve y = -2x^2 + 2. Label its width as dx and its height as y or f(x). 4.3.2 Determine, by using integration, the magnitude of the area indicated in QUESTION 4.3.1. Step 1: Set up the definite integral for the area. The area A under the curve y = -2x^2 + 2 from x = -1 to x = 1 is given by the integral: A = _-1^1 (-2x^2 + 2)\,dx Step 2: Integrate the function. A = [ -(2x^3)/(3) + 2x ]_-1^1 Step 3: Evaluate the definite integral using the Fundamental Theorem of Calculus. A = ( -(2(1)^3)/(3) + 2(1) ) - ( -(2(-1)^3)/(3) + 2(-1) ) A = ( -(2)/(3) + 2 ) - ( -(2(-1))/(3) - 2 ) A = ( -(2)/(3) + (6)/(3) ) - ( (2)/(3) - (6)/(3) ) A = ( (4)/(3) ) - ( -(4)/(3) ) A = (4)/(3) + (4)/(3) A = (8)/(3) The magnitude of the area is (8)/(3) square units. What's next? 📸

Accepted Answer · 2026-07-06T06:10:44.898Z

here are the solutions for the area calculation. QUESTION 4.3 4.3.1 Sketch and clearly indicate the area enclosed by the graph y = -2x^2 + 2, the lines x = -1 and x = 1. Also clearly show the representative strip that will be used to calculate the area enclosed. Step 1: Identify the type of graph and key points. The equation y = -2x^2 + 2 represents a parabola opening downwards. • The y-intercept is found by setting x=0: y = -2(0)^2 + 2 = 2. So, the point is (0, 2). This is also the vertex. • The x-intercepts are found by setting y=0: 0 = -2x^2 + 2 2x^2 = 2 x^2 = 1 x = ± 1. So, the points are (-1, 0) and (1, 0). Step 2: Describe the sketch. Draw a coordinate plane. Plot the vertex at (0, 2) and the x-intercepts at (-1, 0) and (1, 0). Draw a smooth parabola opening downwards through these points. The lines x = -1 and x = 1 are vertical lines that pass through the x-intercepts. The area enclosed is the region under the parabola and above the x-axis, between x=-1 and x=1. To show a representative strip, draw a thin vertical rectangle within this enclosed area, from the x-axis up to the curve y = -2x^2 + 2. Label its width as dx and its height as y or f(x). 4.3.2 Determine, by using integration, the magnitude of the area indicated in QUESTION 4.3.1. Step 1: Set up the definite integral for the area. The area A under the curve y = -2x^2 + 2 from x = -1 to x = 1 is given by the integral: A = _-1^1 (-2x^2 + 2)\,dx Step 2: Integrate the function. A = [ -(2x^3)/(3) + 2x ]_-1^1 Step 3: Evaluate the definite integral using the Fundamental Theorem of Calculus. A = ( -(2(1)^3)/(3) + 2(1) ) - ( -(2(-1)^3)/(3) + 2(-1) ) A = ( -(2)/(3) + 2 ) - ( -(2(-1))/(3) - 2 ) A = ( -(2)/(3) + (6)/(3) ) - ( (2)/(3) - (6)/(3) ) A = ( (4)/(3) ) - ( -(4)/(3) ) A = (4)/(3) + (4)/(3) A = (8)/(3) The magnitude of the area is (8)/(3) square units. What's next? 📸

Sketch and clearly indicate the area enclosed by the graph y = -2x² + 2 the lines x = -1 and x = 1. Also clearly show the representative strip that will be used to calculate the area enclosed.

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Sketch and clearly indicate the area enclosed by the graph y = -2x² + 2 the lines x = -1 and x = 1. Also clearly show the representative strip that will be used to calculate the area enclosed.

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