Final answer:
To find the area bounded by the curves y=1-x² and y=3-3x, we need to find the points of intersection and integrate the difference between the upper and lower curves. The area bounded by the curves is 25/6 square units.
Step-by-step explanation:
To find the area bounded by the curves y=1-x² and y=3-3x, we need to find the points where the curves intersect. Setting the two equations equal to each other, we get (1-x²) = (3-3x). Simplifying, we get x²-3x-2 = 0. Factoring, we get (x-2)(x+1) = 0. So x = 2 and x = -1. Now we can find the y-values by plugging these x-values into either of the equations. Plugging x = 2 into y=1-x², we get y = 1-2² = -3. Plugging x = -1 into y=1-x², we get y = 1-(-1)² = 0. Therefore, the area bounded by the curves is the integral of the upper curve minus the lower curve from x = -1 to x = 2. Integrating y=1-x² gives the function F(x) = x-x³/3 and integrating y=3-3x gives the function G(x) = 3x-3x²/2. So the area is given by the equation A = G(x) - F(x), where A is the area, G(x) is the integral of the upper curve, and F(x) is the integral of the lower curve. Plugging in the values of x, we get A = G(2) - F(2) - (G(-1) - F(-1)). Evaluating the functions at these points, we get A = (3(2)-3(2)²/2) - (2-2³/3) - (3(-1)-3(-1)²/2) + (-1-(-1)³/3). Simplifying, we get A = 3 - 4/3 + 3/2 + 5/3 = 25/6. Therefore, the area bounded by the curves is 25/6 square units.