A glass artist designs a window panel. Seen from the front, the lead outline follows the curve y =…

Step 1. The zeros. $x(x - 4)(x + 1) = 0$ at $x = -1$, $0$ and $4$. The small region lies between $-1$ and $0$, and the large region lies between $0$ and $4$.

Step 2. The antiderivative. $F(x) = \dfrac{x^4}{4} - x^3 - 2x^2$. Then $F(-1) = \dfrac{1}{4} + 1 - 2 = -\dfrac{3}{4}$, $F(0) = 0$ and $F(4) = 64 - 64 - 32 = -32$.

Step 3. The small region. The integral from $-1$ to $0$ is $0 - \left(-\dfrac{3}{4}\right) = \dfrac{3}{4}$, which is positive. The area is $0.75$ square decimetres.

Step 4. The large region. The integral from $0$ to $4$ is $-32 - 0 = -32$, so the curve is below the axis. The area is $32$ square decimetres.

Step 5. The cost. Red glass: $0.75 \times 120 = 90$ rupees. Blue glass: $32 \times 15 = 480$ rupees. The total is $570$ rupees.