Find the area of the region bounded by the line y = 2x + 3, the x-axis and the lines x = −3 and x = 1.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. Line y = 2x + 3, the x-axis and the lines x = −3 and x = 1. To find: the area of the region they enclose.
2 · Draw the figure. Draw the line through (−3/2, 0) and (0, 3). Draw x = −3 and x = 1 as dotted lines. Shade the small triangle below the x-axis on the left and the larger one above it on the right.
3 · Write the formula, rule or theorem. The line meets the x-axis where 2x + 3 = 0, at x = −3/2. Split there: Area = |∫ (2x + 3) dx from −3 to −3/2| + ∫ (2x + 3) dx from −3/2 to 1
4 · Evaluate each integral.
∫ (2x + 3) dx = x² + 3x
From −3 to −3/2: (9/4 − 9/2) − (9 − 9) = −9/4, so this part has area 9/4
From −3/2 to 1: (1 + 3) − (−9/4) = 25/4
5 · Add the parts. Area = 9/4 + 25/4 = 34/4 = 17/2
6 · Check and box the answer. Check with triangles: (1/2) × (3/2) × 3 = 9/4 and (1/2) × (5/2) × 5 = 25/4. Area = 17/2 sq units.
Scoring tip: Find where the line cuts the x-axis first. Integrating straight from −3 to 1 gives 4, which is not the area.
