Working with area functions Consider the function ƒ and the points a, b, and c.
(c) Evaluate A(b) and A(c). Interpret the results using the graphs of part (b) .
ƒ(𝓍) = ― 12𝓍 (𝓍―1) (𝓍― 2) ; a = 0 , b = 1 , c = 2

Verified step by step guidance

1. Step 1: Understand the problem. The function ƒ(𝓍) = -12𝓍(𝓍 - 1)(𝓍 - 2) is given, and we are tasked with evaluating A(b) and A(c), where A(x) represents the area under the curve of ƒ(𝓍) from a = 0 to x. This involves calculating definite integrals of ƒ(𝓍) over the intervals [0, b] and [0, c].
2. Step 2: Set up the integral for A(b). To find A(b), compute the definite integral of ƒ(𝓍) from 0 to b. The integral expression is: ${\int }_0^b-12x(x-1)(x-2)dx$.
3. Step 3: Set up the integral for A(c). Similarly, to find A(c), compute the definite integral of ƒ(𝓍) from 0 to c. The integral expression is: ${\int }_0^c-12x(x-1)(x-2)dx$.
4. Step 4: Interpret the results graphically. The values of A(b) and A(c) represent the signed areas under the curve of ƒ(𝓍) from x = 0 to x = b and x = 0 to x = c, respectively. Positive areas correspond to regions above the x-axis, while negative areas correspond to regions below the x-axis. Use the graph of ƒ(𝓍) to visualize these areas.
5. Step 5: Solve the integrals. To evaluate A(b) and A(c), expand the polynomial -12𝓍(𝓍 - 1)(𝓍 - 2), integrate term by term, and apply the limits of integration. This step involves algebraic manipulation and the fundamental theorem of calculus.

Key Concepts

Area Function

An area function, often denoted as A(x), represents the accumulated area under a curve from a starting point to a variable endpoint x. In this context, A(b) and A(c) will provide the total area under the curve of the function ƒ(x) from the point a to points b and c, respectively. Understanding how to compute and interpret these areas is crucial for analyzing the behavior of the function over specified intervals.

Definite Integral

The definite integral of a function over an interval gives the net area between the function and the x-axis within that interval. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. Evaluating A(b) and A(c) involves computing the definite integral of ƒ(x) from a to b and from a to c, respectively, which quantifies the area under the curve for those segments.

Graphical Interpretation

Graphical interpretation involves analyzing the visual representation of a function to understand its behavior and the significance of calculated areas. By examining the graphs from part (b), one can see how the areas A(b) and A(c) correspond to the regions under the curve of ƒ(x) between the specified points. This helps in understanding the relationship between the algebraic results of the area function and their geometric implications.