Application of IntegralsClass 12 Mathematics Case Study Questions

A civil engineer is designing a bridge to span a river. The main supporting arch of the bridge is in the shape of a parabola with its vertex at the highest point. The equation of this parabolic arch is modeled as $y = -\frac{1}{40}x^2 + 10$, where $y$ represents the height of the arch in meters from the horizontal roadbed, and $x$ is the horizontal distance in meters from the center of the bridge. The roadbed corresponds to the x-axis ($y=0$). To estimate the amount of paint required for the side of the arch, the engineer needs to calculate the area of the region bounded by the parabola and the roadbed.

A graphic designer is creating a logo for a new tech company. The design is based on the area enclosed by two simple curves. The logo's shape is the region bounded by the upward-opening parabola $y = x^2$ and the straight line $y = x + 2$. The dimensions are measured in centimeters. To calculate the material cost for a metal version of the logo, the designer must first determine its exact area.

A city is planning a new botanical garden in the shape of an ellipse. The equation for the boundary of the garden is given by $\frac{x^2}{2500} + \frac{y^2}{1600} = 1$, with all measurements in meters. The garden will be covered entirely with grass. The city planners need to calculate the total area of the garden to order the right amount of turf. They will use the standard formula for the area of an ellipse, which is derived using integration.