[math]\boldsymbol{A} = \int_0^4 { \left\lfloor x^2+1\right\rfloor \mathrm{d} x} = \\[br]\quad = \left(\sqrt{1} - \sqrt{0} \right) \cdot 1 + \left(\sqrt{2} - \sqrt{1} \right) \cdot 2 [br]+ \left(\sqrt{3} - \sqrt{2} \right) \cdot 3 + ... + \left(\sqrt{16} - \sqrt{15} \right) \cdot 16 =\\[br]\quad = 1 \cdot \sqrt{1} + 2 \cdot \sqrt{2} - 2 \cdot \sqrt{1} +3 \cdot \sqrt{3} - 3 \cdot \sqrt{2} + 4 \cdot \sqrt{4} - 4 \cdot \sqrt{3} + ... + 15 \cdot \sqrt{15} - 15 \cdot \sqrt{14} + 16 \cdot \sqrt{16} - 16 \cdot \sqrt{15} = \\[br]\quad =-\sqrt{1} - \sqrt{2} - \sqrt{3} - \sqrt{4} - ... -\sqrt{15} + 16 \cdot \sqrt{16} =\\[br]\quad = - \sum_{i=1}^{15} {\sqrt{i} + 64 \approx \boldsymbol{23,53}[br][/math]