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A trapezoid is a quadrilateral with at least one pair of parallel sides. To understand any shape, especially in coordinate geometry, knowing the vertices is crucial. The vertices of a trapezoid define its shape and size. The vertices of our trapezoid are given by the coordinates \((-2, -2), (1, 1), (5, 1), (7, -2)\). This arrangement ensures that the sides between these points form the boundary of the trapezoid.To check if these points form a trapezoid, focus on:

• The parallel nature of two sides. Here, segments from \((1, 1)\) to \((5, 1)\) and from \((-2, -2)\) to \((7, -2)\) are parallel because they share the same y-coordinate.
• Non-parallel sides that connect these parallel segments, forming a closed structure. The sides from \((-2, -2)\) to \((1, 1)\) and from \((5, 1)\) to \((7, -2)\) serve this purpose.

By plotting these points on a coordinate plane, we confirm they form a trapezoid with a distinct shape.

To find the equation of a line in a Cartesian plane, the slope-intercept form is a helpful tool: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope \(m\) is calculated using the coordinates of two points on the line:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]For the line passing through \((-2, -2)\) and \((1, 1)\):\[m = \frac{1 - (-2)}{1 - (-2)} = 1\]Plug in the slope and a point like \((1, 1)\) into \(y = mx + b\) to solve for \(b\):\[1 = 1(1) + b \Rightarrow b = 0\]Thus, the line equation becomes \(y = x\).Similarly, for the line through \((5, 1)\) and \((7, -2)\):\[m = \frac{-2 - 1}{7 - 5} = -\frac{3}{2}\]\[1 = -\frac{3}{2}(5) + b \Rightarrow b = \frac{17}{2}\]So, the line equation is \(y = -\frac{3}{2}x + \frac{17}{2}\). These line equations are essential in setting up integrals for area calculations.

Integrals can be a powerful tool for finding areas under curves. The definite integral represents the signed area of a curve between two bounds. In this case, we are finding the area between two lines over a specific interval, calculated through:\[\int_{a}^{b} f(x) \, dx\]For our trapezoid, we need the integral from \(-2\) to \(7\) of the function resulting from subtracting the two line equations. This subtraction gives the vertical distance (height) between the lines, which we integrate to find the entire area.The integral setup here is:\[\int_{-2}^{7} \left(\frac{17}{2} - \frac{3}{2}x - x\right) \, dx = \int_{-2}^{7} \left(\frac{17}{2} - \frac{5}{2}x\right) \, dx\]Evaluating this integral involves calculating the antiderivative and replacing \(x\) by upper and lower limits. This gives the area trapped between these lines. The final calculation involves substituting and simplifying to reveal that the area equals 36 square units.

Combining geometry and calculus provides a comprehensive method for solving complex area problems. The integration of geometry with calculus is evident in how we found the area of the trapezoid. Using geometry, we first understood the shape, and via calculus, used definite integrals to calculate the area under curves that form part of a trapezoid. Main steps include:

• Identifying the vertices that form the trapezoid shape.
• Using geometry to establish the necessary line equations.
• Setting up and solving a definite integral that provides the precise area by considering boundaries marked by vertices.

This process of integrating geometric shapes using calculus allows for dynamic problem-solving in mathematics, ensuring accuracy and efficiency in calculating areas beyond simple shapes.