Word Count : 517

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(a) पतली शीट का गुरुत्व केन्द्र वक्रों से घेरित क्षेत्र: (y = 4x^2) और (x = 4) घनत्व फलन: (\delta(x,y) = y) चरण 1: क्षेत्रफल का द्रव्यमान पतली शीट का द्रव्यमान (M) है: [ M = \iint_R \delta(x,y), dA ] यहाँ (R) वह क्षेत्र है जहाँ (0 \le x \le 4) और (0 \le y \le 4x^2) चूंकि (y) उपरी सीमा है, इसलिए (dA = dy,dx): [ M = \int_0^4 \int_0^{4x^2} y, dy, dx ] चरण 2: (y) के सापेक्ष इंटेग्रल [ \int_0^{4x^2} y, dy = \left[ \frac{y^2}{2} \right]_0^{4x^2} = \frac{(4x^2)^2}{2} = \frac{16x^4}{2} = 8x^4 ] इसलिए [ M = \int_0^4 8x^4, dx _______ _____ ________ _____ ________ _______ ________ _________ _________ _____ _______.
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