Given that \(\displaystyle\int_3^{10}\mathrm{f}(x)\,\mathrm{d}x=15\) and \(\mathrm{f}(x)\) lies above the \(x\)-axis for \(x\ge3\), find the exact value of \(\displaystyle\int_3^7[5x^2-3\mathrm{f}(x)]\,\mathrm{d}x-\int_7^{10}3\mathrm{f}(x)\,\mathrm{d}x\).[3]
The diagram shows part of the graph of \(y=\mathrm{g}(x)\). Given that the curve passes through the points \((0,1)\), \((-1,0)\) and \((2,9)\), find the value of \(\displaystyle\int_0^2y\,\mathrm{d}x+\int_1^9x\,\mathrm{d}y\).[1]
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