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.
ScanSolve AI Answer
To calculate the magnitude of the area bounded by the graph y = -x^3 - 9x^2 + 90, the x-axis, and the lines x = -4 and x = 2, we need to evaluate the definite integral of the function from x = -4 to x = 2. From the graph, we can observe that the function y = -x^3 - 9x^2 + 90 is above the x-axis for the entire interval [-4, 2]. Therefore, the area can be calculated directly by integrating the function. Step 1: Set up the definite integral. A = _-4^2 (-x^3 - 9x^2 + 90)\,dx Step 2: Find the antiderivative of the function. The power rule for integration states that x^n\,dx = x^n+1n+1 + C. (-x^3 - 9x^2 + 90)\,dx = -x^3+13+1 - 9x^2+12+1 + 90x + C = -(x^4)/(4) - 9(x^3)/(3) + 90x + C = -(x^4)/(4) - 3x^3 + 90x + C Step 3: Evaluate the definite integral using the Fundamental Theorem of Calculus. A = [-(x^4)/(4) - 3x^3 + 90x]_-4^2 First, evaluate the antiderivative at the upper limit x=2: F(2) = -((2)^4)/(4) - 3(2)^3 + 90(2) F(2) = -(16)/(4) - 3(8) + 180 F(2) = -4 - 24 + 180 F(2) = 152 Next, evaluate the antiderivative at the lower limit x=-4: F(-4) = -((-4)^4)/(4) - 3(-4)^3 + 90(-4) F(-4) = -(256)/(4) - 3(-64) - 360 F(-4) = -64 + 192 - 360 F(-4) = 128 - 360 F(-4) = -232 Step 4: Subtract the value at the lower limit from the value at the upper limit. A = F(2) - F(-4) A = 152 - (-232) A = 152 + 232 A = 384 The magnitude of the window area is 384 square units. Last free one today — make it count tomorrow, or type /upgrade for unlimited.