How are problems that say “evaluate the integral by interpreting it in terms of areas” intended to be solved?
Generally when you evaluate integrals, you use the Fundamental Theorem of Calculus. But remember the definition of the integral comes from wanting a way to calculate the area between a curve and the x-axis. To evaluate an integral by interpreting it in terms of areas means that you should draw a picture of the function and figure out which region’s area is calculable by the integral. Then, instead of using the Fundamental Theorem of Calculus, you go back to your knowledge of geometry and just figure out what the area of the region is using what you know about polygons, circles, ellipses, etc. Then the area you find will be what the integral evaluates to. Done.
Note: I am glossing over one issue here --> remember that integrating over regions above the x-axis give positive results, but below the x-axis give negative results. So you may have to take that into account when finding the areas of the figures.