Area of a region bounded by curves — with a graph and a step-by-step solution
Lines
Enter the lines as in your problem: y = 5x − x², y = x, vertical lines x = 1, the axis y = 0. You can paste all the lines into one field, separated by commas.
Area
📚 Theory: how to find the area of a region bounded by curves
Step 1. Intersection points. Set the functions equal and solve f(x) = g(x): its roots are the bounds of the region along x. If the lines x = a and x = b are given, the bounds are already known.
Step 2. Which line is on top. Substitute any point between the bounds: the function with the larger value is the upper line. If the lines cross inside the interval, the region splits into parts, each with its own upper line.
Step 3. The integral. The area is the integral of the difference “upper minus lower”: S = ∫ (f(x) − g(x)) dx from a to b. The antiderivative and the Newton–Leibniz formula give the exact answer. Area is always positive: the parts are computed separately and added up.
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